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				<h1 class="entry-title">Модуль random. Часть 4</h1>
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<p class="has-light-gray-color has-text-color"><a href="https://www.dmitrymakarov.ru/">Все курсы</a> &gt; <a href="https://www.dmitrymakarov.ru/python/">Программирование на Питоне</a> &gt; Занятие 11 (часть 4)</p>



<p>Рассмотрим способы расчета комбинаций с помощью Питона. Помимо модуля random изучим модуль itertools.</p>


<div class="wp-block-ub-table-of-contents-block ub_table-of-contents" id="ub_table-of-contents-af3e9276-912f-4552-9a0f-bc98f4a9a5cb" data-linktodivider="false" data-showtext="show" data-hidetext="hide" data-scrolltype="auto" data-initiallyhideonmobile="false"
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                    <div class="ub_table-of-contents-title"><strong>Содержание занятия</strong></div></div></div><div class="ub_table-of-contents-extra-container"><div class="ub_table-of-contents-container ub_table-of-contents-1-column "><ul><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#0-kombinatorika-v-pitone>Комбинаторика в Питоне</a><ul><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#1-nprandomshuffle-i-nprandompermutation>np.random.shuffle() и np.random.permutation()</a></li><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#2-modul-itertools>Модуль itertools</a><ul><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#3-perestanovki>Перестановки</a><ul><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#4-perestanovki-bez-zameny>Перестановки без замены</a><ul><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#5-perestanovki-bez-povtoreniy>Перестановки без повторений</a></li><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#6-razmeshcheniya>Размещения</a></li><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#7-perestanovki-s-povtoreniyami>Перестановки с повторениями</a></li></ul></li><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#8-perestanovki-s-zamenoy>Перестановки с заменой</a></li></ul></li><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#9-sochetaniya>Сочетания</a><ul><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#10-sochetaniya-bez-povtoreniy>Сочетания без повторений</a></li><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#11-sochetaniya-s-povtoreniyami>Сочетания с повторениями</a></li><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#12-binomialnye-koeffitsienty>Биномиальные коэффициенты</a></li></ul></li></ul></li></ul></li><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#13-podvedem-itog>Подведем итог</a><ul><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#14-voprosy-dlya-zakrepleniya>Вопросы для закрепления</a></li><li><a href=https://www.dmitrymakarov.ru/python/random-11-04/#15-otvety-na-voprosy>Ответы на вопросы</a></li></ul></li></ul></div></div></div>


<p class="has-text-align-center has-white-color has-text-color has-background" style="background-color:#004235"><a href="https://colab.research.google.com/drive/1VX5VlmmBXhFqdY4uRxjE1a3uwAqNyNTh?usp=sharing" target="_blank" rel="noreferrer noopener">Продолжим работать в том же блокноте</a>⧉</p>



<h2 class="wp-block-heading" id="0-kombinatorika-v-pitone">Комбинаторика в Питоне</h2>



<h3 class="wp-block-heading" id="1-nprandomshuffle-i-nprandompermutation">np.random.shuffle() и np.random.permutation()</h3>



<p>Обе эти функции перемешивают (shuffle) или переставляют (permute) элементы списка или массива. Основное отличие заключается в том, что&nbsp;<strong>np.random.shuffle()</strong>&nbsp;<em>изменяет</em>&nbsp;исходный массив, а&nbsp;<strong>np.random.permutation()</strong>&nbsp;<em>создает «перемешанную» копию</em>&nbsp;исходного массива:</p>



<div class="wp-block-urvanov-syntax-highlighter-code-block"><!-- Urvanov Syntax Highlighter v2.8.37 -->

		<div id="urvanov-syntax-highlighter-672da52084380882902216" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084380882902216-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084380882902216-2">2</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084380882902216-3">3</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084380882902216-4">4</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084380882902216-5">5</div></div>
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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084380882902216-1"><span class="crayon-c"># создадим массив и передадим его в функцию np.random.shuffle()</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084380882902216-2"><span class="crayon-v">arr</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-v">np</span><span class="crayon-sy">.</span><span class="crayon-k ">array</span><span class="crayon-sy">(</span><span class="crayon-sy">[</span><span class="crayon-cn">1</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">3</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">4</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">5</span><span class="crayon-sy">]</span><span class="crayon-sy">)</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084380882902216-3"><span class="crayon-h"> </span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084380882902216-4"><span class="crayon-c"># сама функция выдала None, исходный массив при этом изменился</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084380882902216-5"><span class="crayon-k ">print</span><span class="crayon-sy">(</span><span class="crayon-v">np</span><span class="crayon-sy">.</span><span class="crayon-k ">random</span><span class="crayon-sy">.</span><span class="crayon-e">shuffle</span><span class="crayon-sy">(</span><span class="crayon-v">arr</span><span class="crayon-sy">)</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">arr</span><span class="crayon-sy">)</span></div></div></td>
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		<div id="urvanov-syntax-highlighter-672da5208438f675614159" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da5208438f675614159-1">1</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da5208438f675614159-1">None [3 2 1 5 4]</div></div></td>
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		<div id="urvanov-syntax-highlighter-672da52084391899485190" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084391899485190-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084391899485190-2">2</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084391899485190-3">3</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084391899485190-4">4</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084391899485190-5">5</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084391899485190-6">6</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084391899485190-1"><span class="crayon-c"># еще раз создадим массив</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084391899485190-2"><span class="crayon-v">arr</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-v">np</span><span class="crayon-sy">.</span><span class="crayon-k ">array</span><span class="crayon-sy">(</span><span class="crayon-sy">[</span><span class="crayon-cn">1</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">3</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">4</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">5</span><span class="crayon-sy">]</span><span class="crayon-sy">)</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084391899485190-3"><span class="crayon-h"> </span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084391899485190-4"><span class="crayon-c"># передав его в np.random.permutation(),</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084391899485190-5"><span class="crayon-c"># мы получим перемешанную копию и исходный массив без изменений</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084391899485190-6"><span class="crayon-v">np</span><span class="crayon-sy">.</span><span class="crayon-k ">random</span><span class="crayon-sy">.</span><span class="crayon-e">permutation</span><span class="crayon-sy">(</span><span class="crayon-v">arr</span><span class="crayon-sy">)</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">arr</span></div></div></td>
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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084394655279411-1">1</div></div>
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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084394655279411-1">(array([2, 4, 3, 1, 5]), array([1, 2, 3, 4, 5]))</div></div></td>
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<p>Перемешивание массива можно использовать, если, например, нужно внести элемент случайности в работу с данными. Давайте создадим аналог функции&nbsp;<strong>train_test_split()</strong>&nbsp;библиотеки sklearn, которой мы пользовались для&nbsp;<a href="https://www.dmitrymakarov.ru/intro/train-test-13/">разделения выборки</a>&nbsp;на обучающую и тестовую части.</p>



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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-2">2</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-3">3</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-4">4</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-5">5</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-6">6</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-7">7</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-8">8</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-9">9</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-10">10</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-11">11</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-12">12</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-13">13</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-14">14</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-15">15</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-16">16</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-17">17</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-18">18</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-19">19</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-20">20</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-21">21</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-22">22</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-23">23</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-24">24</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-25">25</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-26">26</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-27">27</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-28">28</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-29">29</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-30">30</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-31">31</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-32">32</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-33">33</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-34">34</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-35">35</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-36">36</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-37">37</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-38">38</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-39">39</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-40">40</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-41">41</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-42">42</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-43">43</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-44">44</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-45">45</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084395031293647-46">46</div></div>
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<p>Подгрузим данные о пациентах с диабетом из модуля datasets библиотеки sklearn. Как это делается, мы&nbsp;<a href="https://www.dmitrymakarov.ru/intro/regression-14/#1-modul-datasets-biblioteki-scikit-learn">подробно разобрали</a>&nbsp;на занятии по линейной регрессии на вводном курсе.</p>



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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-2">2</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-3">3</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-4">4</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-5">5</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-6">6</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-7">7</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-8">8</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-9">9</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-10">10</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-11">11</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-12">12</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-13">13</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-14">14</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da52084398223644395-15">15</div></div>
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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084398223644395-1"><span class="crayon-r">import</span><span class="crayon-h"> </span><span class="crayon-e">pandas </span><span class="crayon-st">as</span><span class="crayon-h"> </span><span class="crayon-i">pd</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084398223644395-2">&nbsp;</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084398223644395-3"><span class="crayon-c"># импортируем данные о пациентах с диабетом</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084398223644395-4"><span class="crayon-st">from</span><span class="crayon-h"> </span><span class="crayon-v">sklearn</span><span class="crayon-sy">.</span><span class="crayon-e">datasets </span><span class="crayon-r">import</span><span class="crayon-h"> </span><span class="crayon-v">load</span><span class="crayon-sy">_</span>diabetes</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084398223644395-5"><span class="crayon-h"> </span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084398223644395-6"><span class="crayon-c"># помещаем их в переменную data</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084398223644395-7"><span class="crayon-v">data</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-e">load_diabetes</span><span class="crayon-sy">(</span><span class="crayon-sy">)</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084398223644395-8"><span class="crayon-h"> </span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084398223644395-9"><span class="crayon-c"># создаем два датафрейма</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084398223644395-10"><span class="crayon-v">X</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-v">pd</span><span class="crayon-sy">.</span><span class="crayon-e">DataFrame</span><span class="crayon-sy">(</span><span class="crayon-v">data</span><span class="crayon-sy">.</span><span class="crayon-v">data</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">columns</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-v">data</span><span class="crayon-sy">.</span><span class="crayon-v">feature_names</span><span class="crayon-sy">)</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084398223644395-11"><span class="crayon-v">y</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-v">pd</span><span class="crayon-sy">.</span><span class="crayon-e">DataFrame</span><span class="crayon-sy">(</span><span class="crayon-v">data</span><span class="crayon-sy">.</span><span class="crayon-v">target</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">columns</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-sy">[</span><span class="crayon-s">'target'</span><span class="crayon-sy">]</span><span class="crayon-sy">)</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084398223644395-12"><span class="crayon-h"> </span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084398223644395-13"><span class="crayon-c"># для проверки работы функции можно также создать массивы Numpy</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da52084398223644395-14"><span class="crayon-c"># X = data.data</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da52084398223644395-15"><span class="crayon-c"># y = data.target</span></div></div></td>
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<p>Вызовем созданную функцию и проверим ее работу.</p>



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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da5208439b678219698-1"><span class="crayon-v">X_train</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">X_test</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">y_train</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">y_test</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-e">split_data</span><span class="crayon-sy">(</span><span class="crayon-v">X</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">y</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">random_state</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-cn">42</span><span class="crayon-sy">)</span></div></div></td>
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<p>Теперь проверим, правильно ли записались данные в созданные выше переменные и расположены ли в них строки в случайном порядке.</p>



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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da5208439c316572168-1"><span class="crayon-v">X_train</span><span class="crayon-sy">.</span><span class="crayon-e">head</span><span class="crayon-sy">(</span><span class="crayon-cn">3</span><span class="crayon-sy">)</span></div></div></td>
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<figure class="wp-block-image size-large"><img fetchpriority="high" decoding="async" width="1024" height="172" src="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/X_train_load_diabetes-1024x172.jpg" alt="функция для разделения выборки на обучающую и тестовую части" class="wp-image-11792" srcset="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/X_train_load_diabetes-1024x172.jpg 1024w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/X_train_load_diabetes-300x50.jpg 300w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/X_train_load_diabetes-768x129.jpg 768w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/X_train_load_diabetes-1536x258.jpg 1536w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/X_train_load_diabetes-2048x344.jpg 2048w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/X_train_load_diabetes-1200x202.jpg 1200w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/X_train_load_diabetes-600x101.jpg 600w" sizes="(max-width: 1024px) 100vw, 1024px" /></figure>



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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da5208439e829413166-1"><span class="crayon-v">y_train</span><span class="crayon-sy">.</span><span class="crayon-e">head</span><span class="crayon-sy">(</span><span class="crayon-cn">3</span><span class="crayon-sy">)</span></div></div></td>
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<figure class="wp-block-image size-full is-resized"><img decoding="async" width="309" height="390" src="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/y_train_load_diabetes.jpg" alt="функция для разделения выборки на обучающую и тестовую части 2" class="wp-image-11793" style="width:123px;height:155px" srcset="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/y_train_load_diabetes.jpg 309w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/y_train_load_diabetes-238x300.jpg 238w" sizes="(max-width: 309px) 100vw, 309px" /></figure>



<p>Похоже, что все работает как надо.</p>



<h3 class="wp-block-heading" id="2-modul-itertools">Модуль itertools</h3>



<p>Рассмотренные выше функции не позволяют вывести все возможные комбинации элементов. Для этого есть модуль&nbsp;<strong>itertools</strong>.</p>



<h4 class="wp-block-heading" id="3-perestanovki">Перестановки</h4>



<p>Напомню, что перестановка (permutation) в комбинаторике позволяет узнать, сколькими способами можно разместить элементы множества в определенном порядке.</p>



<h5 class="wp-block-heading" id="4-perestanovki-bez-zameny">Перестановки без замены</h5>



<p>Вначале возьмем те случаи, когда один элемент может повторяться только один раз. В этом случае мы говорим о&nbsp;<strong>перестановках без замены</strong>&nbsp;(permutations without replacement). Они бывают двух видов: без повторений и с повторениями.</p>



<h6 class="wp-block-heading" id="5-perestanovki-bez-povtoreniy">Перестановки без повторений</h6>



<p>Рассмотрим простой пример. У нас есть три элемента A, B и C. Сколькими способами можно переставить эти объекты так, чтобы порядок элементов не повторялся?</p>



<p class="has-background" style="background-color:#ffe8c4">Так как элементы в исходном множестве не повторяются, то такая перестановка называется&nbsp;<strong>перестановкой без повторения</strong>&nbsp;(permutation without repetition).</p>



<p>Очевидно, на первое место мы можем поставить любой из трех элементов, на второе &#8212; только один из оставшихся двух, на третьем месте окажется единственный не выбывший элемент. Получается $ 3 \times 2 \times 1 = 6 $. Такой расчет не что иное, как факториал числа три, $3!$.</p>



<p>$$ P(n) = n! \rightarrow P(3) = 3! = 6 $$</p>



<p>На Питоне мы можем вычислить факториал нужного нам числа через&nbsp;<strong>функцию factorial()</strong>&nbsp;модуля math.</p>



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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843a0886878035-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843a0886878035-2">2</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843a0886878035-3">3</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843a0886878035-4">4</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843a0886878035-5">5</div></div>
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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843a0886878035-1"><span class="crayon-c"># импортируем модуль math</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843a0886878035-2"><span class="crayon-r">import</span><span class="crayon-h"> </span><span class="crayon-k ">math</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843a0886878035-3"><span class="crayon-h"> </span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843a0886878035-4"><span class="crayon-c"># передадим функции factorial() число 3</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843a0886878035-5"><span class="crayon-k ">math</span><span class="crayon-sy">.</span><span class="crayon-e">factorial</span><span class="crayon-sy">(</span><span class="crayon-cn">3</span><span class="crayon-sy">)</span></div></div></td>
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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843a6174061883-1">6</div></div></td>
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<p>Кроме того, мы можем воспользоваться&nbsp;<strong>функцией permutations()</strong>&nbsp;модуля itertools. В этом случае мы можем вывести все возможные перестановки элементов множества.</p>



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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843a8997875127-1"><span class="crayon-c"># импортируем модуль itertools</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843a8997875127-2"><span class="crayon-r">import</span><span class="crayon-h"> </span><span class="crayon-k ">itertools</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843a8997875127-3"><span class="crayon-h"> </span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843a8997875127-4"><span class="crayon-c"># создадим строку из букв A, B, C</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843a8997875127-5"><span class="crayon-v">x</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-s">'ABC'</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843a8997875127-6"><span class="crayon-c"># помимо строки можно использовать и список</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843a8997875127-7"><span class="crayon-c"># x = ['A', 'B', 'C']</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843a8997875127-8"><span class="crayon-h"> </span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843a8997875127-9"><span class="crayon-c"># и передадим ее в функцию permutations()</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843a8997875127-10"><span class="crayon-c"># так как функция возвращает объект itertools.permutations,</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843a8997875127-11"><span class="crayon-c"># для вывода результата используем функцию list()</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843a8997875127-12"><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">permutations</span><span class="crayon-sy">(</span><span class="crayon-v">x</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div></div></td>
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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843aa956037243-1">[('A', 'B', 'C'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843aa956037243-2"> ('A', 'C', 'B'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843aa956037243-3"> ('B', 'A', 'C'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843aa956037243-4"> ('B', 'C', 'A'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843aa956037243-5"> ('C', 'A', 'B'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843aa956037243-6"> ('C', 'B', 'A')]</div></div></td>
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<p>Если все же нас интересуют не сами перестановки, а их количество, можно воспользоваться функцией&nbsp;<strong>len()</strong>.</p>



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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843ac808834441-1"><span class="crayon-k ">len</span><span class="crayon-sy">(</span><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">permutations</span><span class="crayon-sy">(</span><span class="crayon-v">x</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div></div></td>
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<h6 class="wp-block-heading" id="6-razmeshcheniya">Размещения</h6>



<p>Приведем чуть более сложный пример. Предположим, в соревнованиях участвуют шесть команд ($n = 6$), и нам нужно узнать, сколькими способами эти команды могут занять первое, второе и третье призовые места. Просто вычислить факториал мы уже не можем, нам нужно уменьшить общее число вариантов на то количество перестановок, которое не будет реализовано, потому что мест всего три ($r = 3$).</p>



<p>Такой тип перестановок называют <strong>размещением</strong> (partial permulation). Приведем формулу.</p>



<p>$$ P(n, r) = \frac{n!}{(n-r)!} \rightarrow P(6, 3) = \frac{6!}{(6-3)!} = 120 $$</p>



<p>На Питоне мы можем применить ту же функцию&nbsp;<strong>permutations()</strong>, передав ей через параметр $r$ размер выборки (количество мест).</p>



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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843b0995341029-1"><span class="crayon-c"># теперь элементов исходного множества шесть</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843b0995341029-2"><span class="crayon-v">x</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-s">'ABCDEF'</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843b0995341029-3"><span class="crayon-h"> </span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843b0995341029-4"><span class="crayon-c"># чтобы узнать, сколькими способами их можно разместить на трех местах,</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843b0995341029-5"><span class="crayon-c"># передадим параметр r = 3 и выведем первые пять элементов</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843b0995341029-6"><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">permutations</span><span class="crayon-sy">(</span><span class="crayon-v">x</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">r</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-cn">3</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span><span class="crayon-sy">[</span><span class="crayon-o">:</span><span class="crayon-cn">5</span><span class="crayon-sy">]</span></div></div></td>
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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843b2777648645-1">[('A', 'B', 'C'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843b2777648645-2"> ('A', 'B', 'D'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843b2777648645-3"> ('A', 'B', 'E'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843b2777648645-4"> ('A', 'B', 'F'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843b2777648645-5"> ('A', 'C', 'B')]</div></div></td>
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<p>Посмотрим на общее количество перестановок.</p>



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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843b4319982951-1"><span class="crayon-c"># как и в предыдущем случае, воспользуемся функцией len()</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843b4319982951-2"><span class="crayon-k ">len</span><span class="crayon-sy">(</span><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">permutations</span><span class="crayon-sy">(</span><span class="crayon-v">x</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">r</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-cn">3</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div></div></td>
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<p>Дополнительно замечу, что в случае если выборка $r$ также велика, как и множество $n$ (первый пример), то формулу размещения можно записать как</p>



<p>$$ P(n, n) = \frac{n!}{(n-n)!} $$</p>



<p>В знаменателе получается $0!$, и для того чтобы избежать деления на ноль, принято считать, что факториал нуля равен единице, $0! = 1.$</p>



<h6 class="wp-block-heading" id="7-perestanovki-s-povtoreniyami">Перестановки с повторениями</h6>



<p>Давайте дополнительно усложним задачу и рассчитаем количество перестановок букв в слове «молоко». В данном случае мы не можем воспользоваться факториалом, потому что буква «о» повторяется трижды, и соответственно мы можем разместить ее $3!$ способами. Как следствие, для получения правильного ответа нам нужно разделить количество перестановок на шесть, $3! = 6.$</p>



<p class="has-background" style="background-color:#ffe8c4">В таком случае мы говорим про&nbsp;<strong>перестановки с повторениями</strong>&nbsp;(permutations with repetitions).</p>



<p>$$ \frac{6!}{3!} = 120 $$</p>



<p>В целом, если какой-то из элементов повторяется, то общее число перестановок нужно разделить на произведение факториалов повторяющихся элементов (правильнее сказать на произведение факториалов повторяемости каждого элемента, однако факториал неповторяющихся элементов равен $1! = 1$, поэтому ими можно пренебречь).</p>



<p>$$ \frac{P(n, r)}{n_1! \cdot n_2! \cdot &#8230; \cdot n_r!}, \text{где } n_1! + n_2! + &#8230; + n_r! = n $$</p>



<p>Для расчета перестановок с повторениями напишем&nbsp;<a href="https://www.dmitrymakarov.ru/python/functions-06/#3-sobstvennye-funktsii-v-pitone">собственную функцию</a>.</p>



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<p>Рассчитаем количество перестановок с повторениями букв в слове «молоко».</p>



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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843bb693168603-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843bb693168603-2">2</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843bb693168603-3">3</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843bb693168603-4">4</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843bb693168603-5">5</div></div>
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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843bb693168603-1"><span class="crayon-c"># создадим строку со словом "молоко"</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843bb693168603-2"><span class="crayon-v">x</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-s">'МОЛОКО'</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843bb693168603-3"><span class="crayon-h"> </span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843bb693168603-4"><span class="crayon-c"># вызовем функцию</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843bb693168603-5"><span class="crayon-e">permutations_w_repetition</span><span class="crayon-sy">(</span><span class="crayon-v">x</span><span class="crayon-sy">)</span></div></div></td>
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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843bd212244916-1">1</div></div>
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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843bd212244916-1">120</div></div></td>
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<h5 class="wp-block-heading" id="8-perestanovki-s-zamenoy">Перестановки с заменой</h5>



<p><strong>Перестановки с заменой</strong>&nbsp;(permutations with replacement), когда элементы могут использоваться более одного раза, есть не что иное, как прямое или декартово произведение множества само на себя.</p>



<p>Вначале рассмотрим суть декартова произведения.</p>



<p class="has-background" style="background-color:#ffe8c4"><strong>Декартово произведение</strong>&nbsp;(Cartesian product) двух множеств предполагает, что мы создаем все возможные упорядоченные пары исходных множеств.</p>



<p>В частности, если у нас есть одно множество $A$ с элементами $\{ 1, 2, 3 \}$ и множество $B$ с элементами $ \{ x, y \} $, то их произведением $ A \times B $ будут все возможные упорядоченные комбинации этих элементов.</p>



<figure class="wp-block-image size-full"><img decoding="async" width="768" height="192" src="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/cartesian-product.png" alt="декартово произведение" class="wp-image-11881" srcset="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/cartesian-product.png 768w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/cartesian-product-300x75.png 300w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/cartesian-product-600x150.png 600w" sizes="(max-width: 768px) 100vw, 768px" /></figure>



<p>При умножении множества само на себя один раз мы говорим про&nbsp;<strong>квадрат Декарта</strong>&nbsp;(Cartesian square). В частности, если умножить множество действительных чисел $ \mathbb {R} $ само на себя ($ \mathbb{R} \times \mathbb{R} $), то получится&nbsp;<strong>координатная плоскость или декартова система координат</strong>&nbsp;(Cartesian plane).</p>



<figure class="wp-block-image size-full is-resized"><img loading="lazy" decoding="async" width="894" height="894" src="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/cartesian-plane.png" alt="координатная плоскость или декартова система координат" class="wp-image-11878" style="width:394px;height:394px" srcset="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/cartesian-plane.png 894w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/cartesian-plane-300x300.png 300w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/cartesian-plane-150x150.png 150w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/cartesian-plane-768x768.png 768w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/cartesian-plane-600x600.png 600w" sizes="(max-width: 894px) 100vw, 894px" /></figure>



<p class="has-background" style="background-color:#ffe8c4">Замечу, что множество можно умножить само на себя много раз, в этом случае говорят про&nbsp;<strong>декартову степень</strong>&nbsp;(Cartesian power).</p>



<p>При этом оказывается, что перестановка с заменой и есть декартова степень исходного множества.</p>



<p>Например, представим, что в кафе есть два сорта мороженого, ванильное и клубничное, и мы можем заказать любые два шарика. В частности, шарик ванильного или шарик клубничного. Тогда возможные варианты заказа можно представить следующим образом.</p>



<figure class="wp-block-image size-full is-resized"><img loading="lazy" decoding="async" width="756" height="180" src="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/permutations_w_replacement-1.png" alt="возможные варианты заказа" class="wp-image-11887" style="width:592px;height:141px" srcset="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/permutations_w_replacement-1.png 756w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/permutations_w_replacement-1-300x71.png 300w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/permutations_w_replacement-1-600x143.png 600w" sizes="(max-width: 756px) 100vw, 756px" /></figure>



<p>При этом так как это перестановки и порядок имеет значение, <em>(клубника, ваниль)</em> и <em>(ваниль, клубника)</em> это разные заказы.</p>



<p>В модуле itertools для такой операции есть&nbsp;<strong>функция product()</strong>.</p>



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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843bf420983479-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843bf420983479-2">2</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843bf420983479-1"><span class="crayon-c"># посмотрим, сколькими способами можно выбрать два сорта мороженого</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843bf420983479-2"><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">product</span><span class="crayon-sy">(</span><span class="crayon-sy">[</span><span class="crayon-s">'Ваниль'</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-s">'Клубника'</span><span class="crayon-sy">]</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">repeat</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div></div></td>
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				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c3721545219-1">[('Ваниль', 'Ваниль'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843c3721545219-2"> ('Ваниль', 'Клубника'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c3721545219-3"> ('Клубника', 'Ваниль'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843c3721545219-4"> ('Клубника', 'Клубника')]</div></div></td>
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<p>В целом формула прямого произведения выглядит как $n^r$. Например, если во множестве четыре элемента, то поставить два элемента из этих четырех можно шестнадцатью способами ($4^2=16$).</p>



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		<div id="urvanov-syntax-highlighter-672da520843c5840147758" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843c5840147758-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843c5840147758-2">2</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c5840147758-1"><span class="crayon-c"># посмотрим на способы переставить с заменой два элемента из четырех</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843c5840147758-2"><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">product</span><span class="crayon-sy">(</span><span class="crayon-s">'ABCD'</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">repeat</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div></div></td>
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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-2">2</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-3">3</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-4">4</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-5">5</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-6">6</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-7">7</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-8">8</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-9">9</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-10">10</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-11">11</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-12">12</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-13">13</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-14">14</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-15">15</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843c7162776255-16">16</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c7162776255-1">[('A', 'A'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843c7162776255-2"> ('A', 'B'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c7162776255-3"> ('A', 'C'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843c7162776255-4"> ('A', 'D'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c7162776255-5"> ('B', 'A'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843c7162776255-6"> ('B', 'B'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c7162776255-7"> ('B', 'C'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843c7162776255-8"> ('B', 'D'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c7162776255-9"> ('C', 'A'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843c7162776255-10"> ('C', 'B'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c7162776255-11"> ('C', 'C'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843c7162776255-12"> ('C', 'D'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c7162776255-13"> ('D', 'A'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843c7162776255-14"> ('D', 'B'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c7162776255-15"> ('D', 'C'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843c7162776255-16"> ('D', 'D')]</div></div></td>
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<p>Убедимся, что таких способов 16.</p>



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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843c9236444232-1"><span class="crayon-k ">len</span><span class="crayon-sy">(</span><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">product</span><span class="crayon-sy">(</span><span class="crayon-s">'ABCD'</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">repeat</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div></div></td>
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<p>В данном случае мы умножаем множество само на себя два раза.</p>



<figure class="wp-block-image size-full is-resized"><img loading="lazy" decoding="async" width="593" height="282" src="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/ABCD-cartesian-product.png" alt="умножение множества само на себя два раза" class="wp-image-11891" style="width:492px;height:234px" srcset="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/ABCD-cartesian-product.png 593w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/ABCD-cartesian-product-300x143.png 300w" sizes="(max-width: 593px) 100vw, 593px" /></figure>



<p>Замечу, что если умножать множество на себя три раза, то графически понадобился бы куб. Резюмируем все изученные способы перестановок на схеме ниже.</p>



<figure class="wp-block-image size-full is-resized"><img loading="lazy" decoding="async" width="942" height="497" src="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/permutations.png" alt="способы перестановок" class="wp-image-11892" style="width:707px;height:373px" srcset="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/permutations.png 942w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/permutations-300x158.png 300w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/permutations-768x405.png 768w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/permutations-600x317.png 600w" sizes="(max-width: 942px) 100vw, 942px" /></figure>



<h4 class="wp-block-heading" id="9-sochetaniya">Сочетания</h4>



<h5 class="wp-block-heading" id="10-sochetaniya-bez-povtoreniy">Сочетания без повторений</h5>



<p>В отличие от перестановок при расчете сочетаний (combinations) порядок значения не имеет. Например, представьте, что в парном теннисе из пяти кандидатов нам нужно собрать команду из двух человек.</p>



<figure class="wp-block-image size-full is-resized"><img loading="lazy" decoding="async" width="445" height="296" src="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/combinations-example.png" alt="сочетания" class="wp-image-11895" style="width:370px;height:246px" srcset="https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/combinations-example.png 445w, https://www.dmitrymakarov.ru/wp-content/uploads/2022/04/combinations-example-300x200.png 300w" sizes="(max-width: 445px) 100vw, 445px" /></figure>



<p>По сравнению с похожей задачей с призовыми местами, нам не важно кто будет первым, а кто вторым. Если считать через перестановки, то получится так</p>



<p>$$ P(5, 2) = \frac{5!}{(5-2)!} = 20 $$</p>



<p>Для наглядности посмотрим на эти комбинации с помощью Питона.</p>



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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843cd348411747-1"><span class="crayon-c"># возьмем пять элементов</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843cd348411747-2"><span class="crayon-v">x</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-s">'ABCDE'</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843cd348411747-3"><span class="crayon-h"> </span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843cd348411747-4"><span class="crayon-c"># и найдем способ переставить два элемента из этих пяти</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843cd348411747-5"><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">permutations</span><span class="crayon-sy">(</span><span class="crayon-v">x</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">r</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div></div></td>
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<p>Однако обратите внимание, некоторые комбинации, если не учитывать порядок, повторяются (например, $AB$ и $BA$). Поэтому сочетаний (то есть перестановок без учета порядка) меньше. Причем меньше на количество перестановок каждого типа, то есть $r!$. В данном случае перестановок каждого типа два, то есть $2! = 2$.</p>



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<p>Выведем формулу.</p>



<p>$$ C(n, r) = \frac{n!}{(n-r)! r!} \rightarrow C(5, 2) = \frac{5!}{(5-2)! 2!} = 10 $$</p>



<p>В Питоне для сочетаний есть удобная&nbsp;<strong>функция combinations()</strong>.</p>



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						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843d5147622357-1"><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">combinations</span><span class="crayon-sy">(</span><span class="crayon-v">x</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div></div></td>
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<div class="wp-block-urvanov-syntax-highlighter-code-block"><!-- Urvanov Syntax Highlighter v2.8.37 -->

		<div id="urvanov-syntax-highlighter-672da520843d7488341807" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
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				<table class="crayon-table">
					<tr class="urvanov-syntax-highlighter-row">
				<td class="crayon-nums " data-settings="show">
					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843d7488341807-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843d7488341807-2">2</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843d7488341807-3">3</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843d7488341807-4">4</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843d7488341807-5">5</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843d7488341807-6">6</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843d7488341807-7">7</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843d7488341807-8">8</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843d7488341807-9">9</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843d7488341807-10">10</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843d7488341807-1">[('A', 'B'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843d7488341807-2"> ('A', 'C'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843d7488341807-3"> ('A', 'D'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843d7488341807-4"> ('A', 'E'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843d7488341807-5"> ('B', 'C'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843d7488341807-6"> ('B', 'D'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843d7488341807-7"> ('B', 'E'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843d7488341807-8"> ('C', 'D'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843d7488341807-9"> ('C', 'E'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843d7488341807-10"> ('D', 'E')]</div></div></td>
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<div class="wp-block-urvanov-syntax-highlighter-code-block"><!-- Urvanov Syntax Highlighter v2.8.37 -->

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			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
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				<table class="crayon-table">
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				<td class="crayon-nums " data-settings="show">
					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843d9234572731-1">1</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843d9234572731-1"><span class="crayon-k ">len</span><span class="crayon-sy">(</span><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">combinations</span><span class="crayon-sy">(</span><span class="crayon-v">x</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div></div></td>
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<div class="wp-block-urvanov-syntax-highlighter-code-block"><!-- Urvanov Syntax Highlighter v2.8.37 -->

		<div id="urvanov-syntax-highlighter-672da520843db960077290" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
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				<table class="crayon-table">
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				<td class="crayon-nums " data-settings="show">
					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843db960077290-1">1</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843db960077290-1">10</div></div></td>
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<h5 class="wp-block-heading" id="11-sochetaniya-s-povtoreniyami">Сочетания с повторениями</h5>



<p>Если допустить повторение элементов, сочетаний станет больше.</p>



<div class="wp-block-urvanov-syntax-highlighter-code-block"><!-- Urvanov Syntax Highlighter v2.8.37 -->

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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843dd319200082-1">1</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843dd319200082-1"><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">combinations_with_replacement</span><span class="crayon-sy">(</span><span class="crayon-s">'AB'</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div></div></td>
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<div class="wp-block-urvanov-syntax-highlighter-code-block"><!-- Urvanov Syntax Highlighter v2.8.37 -->

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			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
			<div class="urvanov-syntax-highlighter-main" style="">
				<table class="crayon-table">
					<tr class="urvanov-syntax-highlighter-row">
				<td class="crayon-nums " data-settings="show">
					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843df817015775-1">1</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843df817015775-1">[('A', 'A'), ('A', 'B'), ('B', 'B')]</div></div></td>
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<p>В целом количество сочетаний с повторениями (combinations with repetitions или иногда with replacement) вычисляется по следующей формуле.</p>



<p>$$ C(n, r)_{rep} = \frac{(n+r-1)!}{(n-r)! r!} $$</p>



<p>В нашем случае,</p>



<p>$$ C(2, 2)_{rep} = \frac{(2+2-1)!}{(2-2)! 2!} = \frac{3!}{2!} = 3 $$</p>



<h5 class="wp-block-heading" id="12-binomialnye-koeffitsienty">Биномиальные коэффициенты</h5>



<p>Давайте вернемся к биномиальному распределению и его функции вероятности. Напомню, что во <a href="https://www.dmitrymakarov.ru/python/random-11-02/#13-primer-s-raznoy-veroyatnostyu">втором примере</a> мы три раза подбрасывали монету, $n = 3$, вероятность выпадения орла составляла, $p = 0{,}7$, а найти мы должны были выпадение двух орлов, $k = 2$.</p>



<p>$$ P(X = k) = C(n, k) p^k (1-p)^{n-k} $$</p>



<p>$$ P(X = 2) = C(3, 2) \times 0{,}7^2 \times (1-0{,}7)^{3-2} = 3 \times 0{,}7^2 \times 0{,}3^1 = 0{,}441 $$</p>



<p>Первая часть этой формулы и есть количество сочетаний при выпадении двух орлов в трех бросках. Для того чтобы вывести дерево вероятностей, мы можем воспользоваться декартовой степенью множества {решка, орел} или {H, T}.</p>



<div class="wp-block-urvanov-syntax-highlighter-code-block"><!-- Urvanov Syntax Highlighter v2.8.37 -->

		<div id="urvanov-syntax-highlighter-672da520843e1994536969" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
			<div class="urvanov-syntax-highlighter-main" style="">
				<table class="crayon-table">
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				<td class="crayon-nums " data-settings="show">
					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843e1994536969-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843e1994536969-2">2</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843e1994536969-1"><span class="crayon-c"># трижды умножим множество {H, T} само на себя</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843e1994536969-2"><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">product</span><span class="crayon-sy">(</span><span class="crayon-s">'HT'</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">repeat</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-cn">3</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div></div></td>
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		<div id="urvanov-syntax-highlighter-672da520843e4450078972" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
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				<table class="crayon-table">
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				<td class="crayon-nums " data-settings="show">
					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843e4450078972-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843e4450078972-2">2</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843e4450078972-3">3</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843e4450078972-4">4</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843e4450078972-5">5</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843e4450078972-6">6</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843e4450078972-7">7</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843e4450078972-8">8</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843e4450078972-1">[('H', 'H', 'H'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843e4450078972-2"> ('H', 'H', 'T'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843e4450078972-3"> ('H', 'T', 'H'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843e4450078972-4"> ('H', 'T', 'T'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843e4450078972-5"> ('T', 'H', 'H'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843e4450078972-6"> ('T', 'H', 'T'),</div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843e4450078972-7"> ('T', 'T', 'H'),</div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843e4450078972-8"> ('T', 'T', 'T')]</div></div></td>
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<p>Количество же сочетаний рассчитаем через функцию&nbsp;<strong>combinations()</strong>.</p>



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		<div id="urvanov-syntax-highlighter-672da520843e6413587060" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
			<div class="urvanov-syntax-highlighter-main" style="">
				<table class="crayon-table">
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				<td class="crayon-nums " data-settings="show">
					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843e6413587060-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843e6413587060-2">2</div><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843e6413587060-3">3</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843e6413587060-1"><span class="crayon-c"># посмотрим, в скольких комбинациях выпадет две решки при трех бросках</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843e6413587060-2"><span class="crayon-v">comb</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-k ">len</span><span class="crayon-sy">(</span><span class="crayon-k ">list</span><span class="crayon-sy">(</span><span class="crayon-k ">itertools</span><span class="crayon-sy">.</span><span class="crayon-e">combinations</span><span class="crayon-sy">(</span><span class="crayon-s">'ABC'</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span></div><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843e6413587060-3"><span class="crayon-v">comb</span></div></div></td>
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		<div id="urvanov-syntax-highlighter-672da520843e8390040628" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
			<div class="urvanov-syntax-highlighter-main" style="">
				<table class="crayon-table">
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				<td class="crayon-nums " data-settings="show">
					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843e8390040628-1">1</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843e8390040628-1">3</div></div></td>
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<p>Полный расчет на Питоне будет выглядеть так.</p>



<div class="wp-block-urvanov-syntax-highlighter-code-block"><!-- Urvanov Syntax Highlighter v2.8.37 -->

		<div id="urvanov-syntax-highlighter-672da520843ea138103533" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
			<div class="urvanov-syntax-highlighter-main" style="">
				<table class="crayon-table">
					<tr class="urvanov-syntax-highlighter-row">
				<td class="crayon-nums " data-settings="show">
					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843ea138103533-1">1</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843ea138103533-1"><span class="crayon-k ">round</span><span class="crayon-sy">(</span><span class="crayon-v">comb</span><span class="crayon-h"> </span><span class="crayon-o">*</span><span class="crayon-h"> </span><span class="crayon-sy">(</span><span class="crayon-cn">0.7</span><span class="crayon-h"> </span><span class="crayon-o">*</span><span class="crayon-o">*</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-h"> </span><span class="crayon-o">*</span><span class="crayon-h"> </span><span class="crayon-sy">(</span><span class="crayon-cn">1</span><span class="crayon-h"> </span><span class="crayon-o">-</span><span class="crayon-h"> </span><span class="crayon-cn">0.7</span><span class="crayon-sy">)</span><span class="crayon-o">*</span><span class="crayon-o">*</span><span class="crayon-h"> </span><span class="crayon-sy">(</span><span class="crayon-cn">3</span><span class="crayon-h"> </span><span class="crayon-o">-</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">)</span><span class="crayon-sy">)</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-cn">3</span><span class="crayon-sy">)</span></div></div></td>
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		<div id="urvanov-syntax-highlighter-672da520843ec133683256" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
			<div class="urvanov-syntax-highlighter-main" style="">
				<table class="crayon-table">
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				<td class="crayon-nums " data-settings="show">
					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843ec133683256-1">1</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843ec133683256-1">0.441</div></div></td>
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<p>Конечно же этот результат гораздо проще получить через библиотеку scipy.</p>



<div class="wp-block-urvanov-syntax-highlighter-code-block"><!-- Urvanov Syntax Highlighter v2.8.37 -->

		<div id="urvanov-syntax-highlighter-672da520843ee632706550" class="urvanov-syntax-highlighter-syntax crayon-theme-familiar-my-copy urvanov-syntax-highlighter-font-consolas urvanov-syntax-highlighter-os-pc print-yes notranslate" data-settings=" minimize scroll-always" style=" margin-top: 15px; margin-bottom: 15px; margin-left: 15px; margin-right: 15px; font-size: 16px !important; line-height: 22px !important;">
		
			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
			<div class="urvanov-syntax-highlighter-main" style="">
				<table class="crayon-table">
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				<td class="crayon-nums " data-settings="show">
					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843ee632706550-1">1</div><div class="crayon-num crayon-striped-num" data-line="urvanov-syntax-highlighter-672da520843ee632706550-2">2</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843ee632706550-1"><span class="crayon-st">from</span><span class="crayon-h"> </span><span class="crayon-v">scipy</span><span class="crayon-sy">.</span><span class="crayon-e">stats </span><span class="crayon-r">import</span><span class="crayon-h"> </span><span class="crayon-e">binom</span></div><div class="crayon-line crayon-striped-line" id="urvanov-syntax-highlighter-672da520843ee632706550-2"><span class="crayon-v">binom</span><span class="crayon-sy">.</span><span class="crayon-e">pmf</span><span class="crayon-sy">(</span><span class="crayon-v">k</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-cn">2</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">n</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-cn">3</span><span class="crayon-sy">,</span><span class="crayon-h"> </span><span class="crayon-v">p</span><span class="crayon-h"> </span><span class="crayon-o">=</span><span class="crayon-h"> </span><span class="crayon-cn">0.7</span><span class="crayon-sy">)</span><span class="crayon-sy">.</span><span class="crayon-k ">round</span><span class="crayon-sy">(</span><span class="crayon-cn">3</span><span class="crayon-sy">)</span></div></div></td>
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			<div class="urvanov-syntax-highlighter-plain-wrap"></div>
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				<table class="crayon-table">
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					<div class="urvanov-syntax-highlighter-nums-content" style="font-size: 16px !important; line-height: 22px !important;"><div class="crayon-num" data-line="urvanov-syntax-highlighter-672da520843f0088337056-1">1</div></div>
				</td>
						<td class="urvanov-syntax-highlighter-code"><div class="crayon-pre" style="font-size: 16px !important; line-height: 22px !important; -moz-tab-size:4; -o-tab-size:4; -webkit-tab-size:4; tab-size:4;"><div class="crayon-line" id="urvanov-syntax-highlighter-672da520843f0088337056-1">0.441</div></div></td>
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<h2 class="wp-block-heading" id="13-podvedem-itog">Подведем итог</h2>



<p>По большому счету мы затронули три важные темы:</p>



<ol class="wp-block-list">
<li>Моделирование и исследование <a href="https://www.dmitrymakarov.ru/python/random-11/">случайных процессов</a> с помощью модуля random;</li>



<li><a href="https://www.dmitrymakarov.ru/python/random-11-02/">Дискретные</a> и <a href="https://www.dmitrymakarov.ru/python/random-11-03/">непрерывные</a> распределения случайных величин; и</li>



<li>Комбинаторику на Питоне.</li>
</ol>



<p>Пройденный материал поможет как в дальнейшем изучении теории, так и в построении более качественных моделей машинного обучения.</p>



<h3 class="wp-block-heading" id="14-voprosy-dlya-zakrepleniya">Вопросы для закрепления</h3>



<p class="has-black-color has-text-color has-background" style="background-color:#ffe8c4">Вопрос. Чем теоретическая вероятность отличается от эмпирической?</p>


<div class="wp-block-ub-content-toggle" id="ub-content-toggle-ece6dba9-b0ac-438f-9a12-a87d5dcc1d45" data-mobilecollapse="true" data-desktopcollapse="true">
<div class="wp-block-ub-content-toggle-accordion" id="ub-content-toggle-panel-block-9b78d45f-e05e-4c47-9d9b-b45950b79e21">
                <div class="wp-block-ub-content-toggle-accordion-title-wrap"" aria-controls="ub-content-toggle-panel-0-ece6dba9-b0ac-438f-9a12-a87d5dcc1d45" tabindex="0">
                    <p class="wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-ece6dba9-b0ac-438f-9a12-a87d5dcc1d45">Посмотреть правильный ответ</p><div class="wp-block-ub-content-toggle-accordion-toggle-wrap left"><span class="wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-math-plus"></span>
                    </div></div><div role="region" aria-expanded="false" class="wp-block-ub-content-toggle-accordion-content-wrap ub-hide" id="ub-content-toggle-panel-0-ece6dba9-b0ac-438f-9a12-a87d5dcc1d45">

<p>Ответ: <strong>теоретическая</strong>&nbsp;вероятность — это соотношение благоприятствующих событию исходов ко всем возможным исходам испытания, эту вероятность мы можем рассчитать;&nbsp;<strong>эмпирическая</strong>&nbsp;вероятность — соотношение наступления события к общему числу испытаний, которое мы наблюдаем в результате эксперимента</p>

</div></div>
</div>


<p class="has-black-color has-text-color has-background" style="background-color:#ffe8c4">Вопрос. Чем отличается функция вероятности (probability mass function, pmf) от плотности вероятности (probability density function, pdf)?</p>


<div class="wp-block-ub-content-toggle" id="ub-content-toggle-13071540-1081-4fc9-bf6b-4322d84042e1" data-mobilecollapse="true" data-desktopcollapse="true">
<div class="wp-block-ub-content-toggle-accordion" id="ub-content-toggle-panel-block-90d807cb-5c48-40da-8b35-effa7476aee6">
                <div class="wp-block-ub-content-toggle-accordion-title-wrap"" aria-controls="ub-content-toggle-panel-0-13071540-1081-4fc9-bf6b-4322d84042e1" tabindex="0">
                    <p class="wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-13071540-1081-4fc9-bf6b-4322d84042e1">Посмотреть правильный ответ</p><div class="wp-block-ub-content-toggle-accordion-toggle-wrap left"><span class="wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-math-plus"></span>
                    </div></div><div role="region" aria-expanded="false" class="wp-block-ub-content-toggle-accordion-content-wrap ub-hide" id="ub-content-toggle-panel-0-13071540-1081-4fc9-bf6b-4322d84042e1">

<p>Ответ: <strong>функция вероятности</strong>&nbsp;задает распределение дискретной случайной величины (discrete random variable), а&nbsp;<strong>плотность вероятности</strong>&nbsp;описывает распределение непрерывной величины (discrete random variable)</p>

</div></div>
</div>


<p class="has-black-color has-text-color has-background" style="background-color:#ffe8c4">Вопрос. Чем сочетания отличаются от перестановок?</p>


<div class="wp-block-ub-content-toggle" id="ub-content-toggle-2dbacd7f-0456-4e60-b350-9610e52c1ef0" data-mobilecollapse="true" data-desktopcollapse="true">
<div class="wp-block-ub-content-toggle-accordion" id="ub-content-toggle-panel-block-bff49097-0323-42e4-afe2-6de3dac8113f">
                <div class="wp-block-ub-content-toggle-accordion-title-wrap"" aria-controls="ub-content-toggle-panel-0-2dbacd7f-0456-4e60-b350-9610e52c1ef0" tabindex="0">
                    <p class="wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-2dbacd7f-0456-4e60-b350-9610e52c1ef0">Посмотреть правильный ответ</p><div class="wp-block-ub-content-toggle-accordion-toggle-wrap left"><span class="wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-math-plus"></span>
                    </div></div><div role="region" aria-expanded="false" class="wp-block-ub-content-toggle-accordion-content-wrap ub-hide" id="ub-content-toggle-panel-0-2dbacd7f-0456-4e60-b350-9610e52c1ef0">

<p>Ответ: при расчете количества перестановок мы учитываем порядок, в котором располагаются элементы, при расчете количества сочетаний порядок не важен</p>

</div></div>
</div>


<p>На следующем занятии мы немного отдохнем от математики и рассмотрим&nbsp;<a href="https://www.dmitrymakarov.ru/python/oop-12/">объектно-ориентированный подход</a>&nbsp;к программированию.</p>


<div id="ub_divider_78bd2283-65a8-4db5-b3de-4fc3517da5c8" class="wp-block-ub-divider ub-divider-orientation-horizontal"><hr class="ub_divider" ></hr></div>


<h3 class="wp-block-heading" id="15-otvety-na-voprosy">Ответы на вопросы</h3>



<p><strong>Вопрос</strong>. Подскажите, а зачем повторно импортировать Numpy, itertools и math перед объявлением собственной функции permutations_w_repetition()? Ведь вы уже импортировали их перед этим.</p>



<p><strong>Ответ</strong>. Я повторно импортировал необходимые библиотеки на случай использования функции в других проектах. Предположим, вы захотите использовать эту функцию в своей работе и поместите ее в отдельный файл. Для того чтобы код сработал, вам сначала понадобится импортировать эти три библиотеки.</p>



<p class="has-background" style="background-color:#ffe8c4">По-английски они называюся <strong>dependency</strong> или <strong>dependencies</strong>, потому что от них зависит выполнение основной программы.</p>



<p>Дополнительно замечу, что Руководство по написанию кода на Питоне (Python Enhancement Proposal или PEP 8)&nbsp;<a href="https://peps.python.org/pep-0008/#imports" target="_blank" rel="noreferrer noopener">рекомендует</a>⧉ делать импорт необходимых библиотек и модулей «после комментариев к модулю и строк документации, и перед объявлением констант», а не внутри функций или классов. Подробнее об этом мы поговорим через одно занятие.</p>


<div id="ub_divider_dd9713ba-314a-4038-a147-0ed557e1b32e" class="wp-block-ub-divider ub-divider-orientation-horizontal"><hr class="ub_divider" ></hr></div>


<p><strong>Вопрос</strong>. Скажите, а чем матожидание отличается от среднего значения?</p>



<p><strong>Ответ</strong>. Матожидание (expected value) — это то среднее значение, которое мы предполагаем/ожидаем получить с учетом известного нам типа распределения. Например, зная, что случайная величина следует биномиальному распределению с параметрами $n = 3$ и $p = 0{,}7$, мы ожидаем наблюдать среднее значение $np = 3 \times 0{,}7 = 2{,}1$. В частности, с помощью <strong>функции scipy.stats.binom.stats()</strong> мы рассчитываем именно матожидание распределения.</p>



<p>Собрав же фактические данные (мы для этого использовали модуль random и функцию <strong>np.random.binomial()</strong>), мы можем посмотреть насколько среднее (mean value) этих данных согласуется с математическим ожиданием. При достаточно большом количестве экспериментов среднее должно стремиться к математическому ожиданию.</p>



<p>Привел пример в <a href="https://colab.research.google.com/drive/1VX5VlmmBXhFqdY4uRxjE1a3uwAqNyNTh#scrollTo=8pV7hhuLT2Hd" target="_blank" rel="noreferrer noopener">конце блокнота</a>⧉.</p>

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