1004 lines
27 KiB
Plaintext
Executable File
1004 lines
27 KiB
Plaintext
Executable File
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "iWFWs9SjJR1o"
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},
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"source": [
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"### Комбинаторика в Питоне"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "Nh4EhvMbJUIa",
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"outputId": "871cbe2f-db64-4ed6-a5bf-7678acdbdafa"
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"None [2 1 3 4 5]\n"
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]
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}
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],
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"source": [
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"import numpy as np\n",
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"# создадим массив и передадим его в функцию np.random.shuffle()\n",
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"arr = np.array([1, 2, 3, 4, 5])\n",
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"\n",
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"# сама функция выдала None, исходный массив при этом изменился\n",
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"print(np.random.shuffle(arr), arr)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "Xrf3CqP2Fg1O",
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"outputId": "fe95126f-2591-496d-a00a-eea6f92b99d0"
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(array([2, 4, 5, 1, 3]), array([1, 2, 3, 4, 5]))"
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]
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},
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"execution_count": 11,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# еще раз создадим массив\n",
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"arr = np.array([1, 2, 3, 4, 5])\n",
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"\n",
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"# передав его в np.random.permutation(),\n",
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"# мы получим перемешанную копию и исходный массив без изменений\n",
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"np.random.permutation(arr), arr"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "CRc0_oP1eO3y"
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},
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"source": [
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"#### Модуль itertools"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "Ib9u0kxzA785"
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},
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"source": [
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"##### Перестановки"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "WdlV4ZrwdliN"
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},
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"source": [
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"###### Перестановки без замены"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "fOdk6SujdoLH"
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},
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"source": [
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"1. Перестановки без повторений"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "xdlxohSLeRUW",
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"outputId": "9e9bfd0e-9a7a-4a41-a1e5-cb17668c5408"
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"6"
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]
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},
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"execution_count": 12,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# импортируем модуль math\n",
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"import math\n",
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"\n",
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"# передадим функции factorial() число 3\n",
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"math.factorial(3)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "S5ceeFMfeNwT",
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"outputId": "679f6eb5-2ef6-4f6a-e748-ea187470de80"
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[('A', 'B', 'C'),\n",
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" ('A', 'C', 'B'),\n",
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" ('B', 'A', 'C'),\n",
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" ('B', 'C', 'A'),\n",
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" ('C', 'A', 'B'),\n",
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" ('C', 'B', 'A')]"
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]
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},
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"execution_count": 13,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# импортируем модуль itertools\n",
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"import itertools\n",
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"\n",
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"# создадим строку из букв A, B, C\n",
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"x = 'ABC'\n",
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"# помимо строки можно использовать и список\n",
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"# x = ['A', 'B', 'C']\n",
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"\n",
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"# и передадим ее в функцию permutations()\n",
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"# так как функция возвращает объект itertools.permutations,\n",
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"# для вывода результата используем функцию list()\n",
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"list(itertools.permutations(x))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "DufioTwtfH5v",
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"outputId": "93bfc07b-856b-4aa4-8d7f-0b82b20e58c4"
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"6"
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]
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},
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"execution_count": 14,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# чтобы узнать количество перестановок, можно использовать функцию len()\n",
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"len(list(itertools.permutations(x)))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "B-J32YZ4fQCL",
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"outputId": "183f8a99-6ab5-4578-c584-ff5fa45332a7"
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[('A', 'B', 'C'),\n",
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" ('A', 'B', 'D'),\n",
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" ('A', 'B', 'E'),\n",
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" ('A', 'B', 'F'),\n",
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" ('A', 'C', 'B')]"
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]
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},
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"execution_count": 15,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# теперь элементов исходного множества шесть\n",
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"x = 'ABCDEF'\n",
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"\n",
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"# чтобы узнать, сколькими способами их можно разместить на трех местах,\n",
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"# передадим параметр r = 3 и выведем первые пять элементов\n",
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"list(itertools.permutations(x, r = 3))[:5]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "NAkXrmpKfzUC",
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"outputId": "c33775a0-0c6b-4cae-e7cb-1961fe254a5a"
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"120"
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]
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},
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"execution_count": 16,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# посмотрим на общее количество таких перестановок\n",
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"len(list(itertools.permutations(x, r = 3)))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "PqRx8pF0dtKm"
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},
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"source": [
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"2. Перестановки с повторениями"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"metadata": {
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"id": "3qQ5QpTFpIJc"
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},
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"outputs": [],
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"source": [
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"# импортируем необходимые библиотеки\n",
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"import itertools\n",
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"import numpy as np\n",
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"import math\n",
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"\n",
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"# объявим функцию permutations_w_repetition(), которая будет принимать два параметра\n",
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"# x - строка, список или массив Numpy\n",
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"# r - количество мест в перестановке, по умолчанию равно количеству элементов в x\n",
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"def permutations_w_repetition(x, r = len(x)):\n",
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"\n",
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" # если передается строка,\n",
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" if isinstance(x, str):\n",
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" # превращаем ее в список\n",
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" x = list(x)\n",
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"\n",
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" # в числителе рассчитаем количество перестановок без повторений\n",
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" numerator = len(list(itertools.permutations(x, r = r)))\n",
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"\n",
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" # для того чтобы рассчитать знаменатель найдем,\n",
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" # сколько раз повторяется каждый из элементов\n",
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" _, counts = np.unique(x, return_counts = True)\n",
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"\n",
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" # объявим переменную для знаменателя\n",
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" denominator = 1\n",
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"\n",
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" # и в цикле будем помещать туда произведение факториалов\n",
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" # повторяющихся элементов\n",
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" for c in counts:\n",
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"\n",
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" # для этого проверим повторяется ли элемент\n",
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" if c > 1:\n",
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"\n",
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" # и если да, умножим знаменатель на факториал повторяющегося элемента\n",
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" denominator *= math.factorial(c)\n",
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"\n",
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" # разделим числитель на знаменатель\n",
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" # деление дает тип float, поэтому используем функцию int(),\n",
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" # чтобы результат был целым числом\n",
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" return int(numerator/denominator)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 18,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "Fq0vyfdg--5m",
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"outputId": "478c5a0b-0768-473c-ff1f-fd48231e0bda"
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},
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||
"outputs": [
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{
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||
"data": {
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"text/plain": [
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"120"
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||
]
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||
},
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"execution_count": 18,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# создадим строку со словом \"молоко\"\n",
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"x = 'МОЛОКО'\n",
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"\n",
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"# вызовем функцию\n",
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"permutations_w_repetition(x)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "GoAiKH8-pIo3"
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},
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"source": [
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"###### Перестановки с заменой"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 19,
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"metadata": {
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"id": "-uevbl8WdvsA",
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"outputId": "b320a8a8-bfed-43b0-b8bf-8f041c6d256f"
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[('Ваниль', 'Ваниль'),\n",
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" ('Ваниль', 'Клубника'),\n",
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" ('Клубника', 'Ваниль'),\n",
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" ('Клубника', 'Клубника')]"
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]
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},
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"execution_count": 19,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# посмотрим, сколькими способами можно выбрать два сорта мороженого\n",
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"list(itertools.product(['Ваниль', 'Клубника'], repeat = 2))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 20,
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"metadata": {
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||
"colab": {
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||
"base_uri": "https://localhost:8080/"
|
||
},
|
||
"id": "9uxp4aeItTED",
|
||
"outputId": "29ce5a18-39fd-4bf3-b3d5-0eb977763122"
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"[('A', 'A'),\n",
|
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" ('A', 'B'),\n",
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" ('A', 'C'),\n",
|
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" ('A', 'D'),\n",
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" ('B', 'A'),\n",
|
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" ('B', 'B'),\n",
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" ('B', 'C'),\n",
|
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" ('B', 'D'),\n",
|
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" ('C', 'A'),\n",
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" ('C', 'B'),\n",
|
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" ('C', 'C'),\n",
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" ('C', 'D'),\n",
|
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" ('D', 'A'),\n",
|
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" ('D', 'B'),\n",
|
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" ('D', 'C'),\n",
|
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" ('D', 'D')]"
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||
]
|
||
},
|
||
"execution_count": 20,
|
||
"metadata": {},
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||
"output_type": "execute_result"
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||
}
|
||
],
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"source": [
|
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"# посмотрим на способы переставить с заменой два элемента из четырех\n",
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"list(itertools.product('ABCD', repeat = 2))"
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]
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},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 21,
|
||
"metadata": {
|
||
"colab": {
|
||
"base_uri": "https://localhost:8080/"
|
||
},
|
||
"id": "LLlYVJ6Cgwc9",
|
||
"outputId": "fe8ce0c1-d717-49c2-bd59-6c8ae45335fc"
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"16"
|
||
]
|
||
},
|
||
"execution_count": 21,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
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"source": [
|
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"# убедимся, что таких способов 16\n",
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"len(list(itertools.product('ABCD', repeat = 2)))"
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]
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},
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||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"id": "9TKsA791BEES"
|
||
},
|
||
"source": [
|
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"##### Сочетания"
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||
]
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||
},
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||
{
|
||
"cell_type": "code",
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||
"execution_count": 22,
|
||
"metadata": {
|
||
"colab": {
|
||
"base_uri": "https://localhost:8080/"
|
||
},
|
||
"id": "tIbK4PNsyVwB",
|
||
"outputId": "09dbb8bf-a98c-4101-d465-2b6a19e88015"
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"[('A', 'B'),\n",
|
||
" ('A', 'C'),\n",
|
||
" ('A', 'D'),\n",
|
||
" ('A', 'E'),\n",
|
||
" ('B', 'A'),\n",
|
||
" ('B', 'C'),\n",
|
||
" ('B', 'D'),\n",
|
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" ('B', 'E'),\n",
|
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" ('C', 'A'),\n",
|
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" ('C', 'B'),\n",
|
||
" ('C', 'D'),\n",
|
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" ('C', 'E'),\n",
|
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" ('D', 'A'),\n",
|
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" ('D', 'B'),\n",
|
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" ('D', 'C'),\n",
|
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" ('D', 'E'),\n",
|
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" ('E', 'A'),\n",
|
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" ('E', 'B'),\n",
|
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" ('E', 'C'),\n",
|
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" ('E', 'D')]"
|
||
]
|
||
},
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||
"execution_count": 22,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"# возьмем пять элементов\n",
|
||
"x = 'ABCDE'\n",
|
||
"\n",
|
||
"# и найдем способ переставить два элемента из этих пяти\n",
|
||
"list(itertools.permutations(x, r = 2))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 23,
|
||
"metadata": {
|
||
"colab": {
|
||
"base_uri": "https://localhost:8080/"
|
||
},
|
||
"id": "9NJLHZVP9xve",
|
||
"outputId": "1ec51fcc-93ee-4cc9-b130-47935e0a1d01"
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"10"
|
||
]
|
||
},
|
||
"execution_count": 23,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"# уменьшим на количество перестановок каждого типа r!\n",
|
||
"int(len(list(itertools.permutations(x, r = 2)))/math.factorial(2))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 24,
|
||
"metadata": {
|
||
"colab": {
|
||
"base_uri": "https://localhost:8080/"
|
||
},
|
||
"id": "imUV1aSA_JI4",
|
||
"outputId": "44cc4c14-993e-492a-8a17-839cc41d0f0f"
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"[('A', 'B'),\n",
|
||
" ('A', 'C'),\n",
|
||
" ('A', 'D'),\n",
|
||
" ('A', 'E'),\n",
|
||
" ('B', 'C'),\n",
|
||
" ('B', 'D'),\n",
|
||
" ('B', 'E'),\n",
|
||
" ('C', 'D'),\n",
|
||
" ('C', 'E'),\n",
|
||
" ('D', 'E')]"
|
||
]
|
||
},
|
||
"execution_count": 24,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"# то же самое можно рассчитать с помощью функции combinations()\n",
|
||
"list(itertools.combinations(x, 2))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 25,
|
||
"metadata": {
|
||
"colab": {
|
||
"base_uri": "https://localhost:8080/"
|
||
},
|
||
"id": "GNuEIKwl_Px1",
|
||
"outputId": "23f3ec43-59a5-4914-f2ad-aae6291ac8ad"
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"10"
|
||
]
|
||
},
|
||
"execution_count": 25,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"# посмотрим на количество сочетаний\n",
|
||
"len(list(itertools.combinations(x, 2)))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"id": "CPpGYzYJMuiv"
|
||
},
|
||
"source": [
|
||
"Сочетания с заменой"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 26,
|
||
"metadata": {
|
||
"colab": {
|
||
"base_uri": "https://localhost:8080/"
|
||
},
|
||
"id": "75SIWA4TMgj1",
|
||
"outputId": "f622aa9c-c089-4352-c2b1-f5f423947286"
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"[('A', 'A'), ('A', 'B'), ('B', 'B')]"
|
||
]
|
||
},
|
||
"execution_count": 26,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"# сколькими способами с заменой можно выбрать два элемента из двух\n",
|
||
"list(itertools.combinations_with_replacement('AB', 2))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 27,
|
||
"metadata": {
|
||
"colab": {
|
||
"base_uri": "https://localhost:8080/"
|
||
},
|
||
"id": "DgCya-oDUIyd",
|
||
"outputId": "f78b7f06-b597-4006-d9cc-fa49fe45f809"
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"[('A', 'B')]"
|
||
]
|
||
},
|
||
"execution_count": 27,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"# очевидно, что без замены есть только один такой способ\n",
|
||
"list(itertools.combinations('AB', 2))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"id": "9xgg_wXLbjuu"
|
||
},
|
||
"source": [
|
||
"Биномиальные коэффициенты"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 28,
|
||
"metadata": {
|
||
"colab": {
|
||
"base_uri": "https://localhost:8080/"
|
||
},
|
||
"id": "6dDgzQTlB7ZF",
|
||
"outputId": "768cf35c-a548-450e-935c-dc84a6533d70"
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"[('H', 'H', 'H'),\n",
|
||
" ('H', 'H', 'T'),\n",
|
||
" ('H', 'T', 'H'),\n",
|
||
" ('H', 'T', 'T'),\n",
|
||
" ('T', 'H', 'H'),\n",
|
||
" ('T', 'H', 'T'),\n",
|
||
" ('T', 'T', 'H'),\n",
|
||
" ('T', 'T', 'T')]"
|
||
]
|
||
},
|
||
"execution_count": 28,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"# дерево вероятностей можно построить с помощью декартовой степени\n",
|
||
"list(itertools.product('HT', repeat = 3))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 29,
|
||
"metadata": {
|
||
"colab": {
|
||
"base_uri": "https://localhost:8080/"
|
||
},
|
||
"id": "NFqzBH7lblyY",
|
||
"outputId": "e2d1b80c-a022-4878-8a5a-111019cdd561"
|
||
},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"3"
|
||
]
|
||
},
|
||
"execution_count": 29,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"# посмотрим, в скольких комбинациях выпадет два орла при трех бросках\n",
|
||
"comb = len(list(itertools.combinations('ABC', 2)))\n",
|
||
"comb"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"id": "q5YrLz3_sffO"
|
||
},
|
||
"source": [
|
||
"### Упражнения"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"**Задание 1**. Реализовать комбинаторные конструкции без использования библиотеки intertools. Проведите сравнительный анализ на быстродействие и объем используемой памяти. Сделайте выводы."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 30,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"def permutations(items: list[int]):\n",
|
||
" if len(items) == 0:\n",
|
||
" yield []\n",
|
||
" else:\n",
|
||
" for i in range(len(items)):\n",
|
||
" for perm in permutations(items[:i] + items[i+1:]):\n",
|
||
" yield [items[i]] + perm\n",
|
||
"\n",
|
||
"def combinations(items: list[int], k: int):\n",
|
||
" if k == 0:\n",
|
||
" yield []\n",
|
||
" elif len(items) == k:\n",
|
||
" yield items\n",
|
||
" else:\n",
|
||
" for comb in combinations(items[1:], k-1):\n",
|
||
" yield [items[0]] + comb\n",
|
||
" for comb in combinations(items[1:], k):\n",
|
||
" yield comb\n",
|
||
"\n",
|
||
"def arrangements(items: list[int], k: int):\n",
|
||
" if k == 0:\n",
|
||
" yield []\n",
|
||
" else:\n",
|
||
" for i in range(len(items)):\n",
|
||
" for arr in arrangements(items[:i] + items[i+1:], k-1):\n",
|
||
" yield [items[i]] + arr"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 31,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"my func\n",
|
||
"\n",
|
||
"===Permutations===\n",
|
||
"Output: [[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 3, 4, 2], [1, 4, 2, 3], [1, 4, 3, 2], [2, 1, 3, 4], [2, 1, 4, 3], [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 1, 3], [2, 4, 3, 1], [3, 1, 2, 4], [3, 1, 4, 2], [3, 2, 1, 4], [3, 2, 4, 1], [3, 4, 1, 2], [3, 4, 2, 1], [4, 1, 2, 3], [4, 1, 3, 2], [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1]]\n",
|
||
"Time: 4.315376281738281e-05\n",
|
||
"===Combinations===\n",
|
||
"Output: [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]\n",
|
||
"Time: 1.0013580322265625e-05\n",
|
||
"===Arrangements===\n",
|
||
"Output: [[1, 2], [1, 3], [1, 4], [2, 1], [2, 3], [2, 4], [3, 1], [3, 2], [3, 4], [4, 1], [4, 2], [4, 3]]\n",
|
||
"Time: 1.4066696166992188e-05\n",
|
||
"\n",
|
||
"itertools\n",
|
||
"\n",
|
||
"===Permutations===\n",
|
||
"Output: [(1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 2, 4), (1, 3, 4, 2), (1, 4, 2, 3), (1, 4, 3, 2), (2, 1, 3, 4), (2, 1, 4, 3), (2, 3, 1, 4), (2, 3, 4, 1), (2, 4, 1, 3), (2, 4, 3, 1), (3, 1, 2, 4), (3, 1, 4, 2), (3, 2, 1, 4), (3, 2, 4, 1), (3, 4, 1, 2), (3, 4, 2, 1), (4, 1, 2, 3), (4, 1, 3, 2), (4, 2, 1, 3), (4, 2, 3, 1), (4, 3, 1, 2), (4, 3, 2, 1)]\n",
|
||
"Time: 2.1219253540039062e-05\n",
|
||
"===Combinations===\n",
|
||
"Output: [(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)]\n",
|
||
"Time: 5.0067901611328125e-06\n",
|
||
"===Arrangements===\n",
|
||
"Output: [(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)]\n",
|
||
"Time: 5.0067901611328125e-06\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"import time\n",
|
||
"\n",
|
||
"\n",
|
||
"def measure(func, *args):\n",
|
||
" start_time = time.time()\n",
|
||
" print(\"Output:\", list(func(*args)))\n",
|
||
" end_time = time.time()\n",
|
||
" print(\"Time:\", end_time - start_time)\n",
|
||
"\n",
|
||
"\n",
|
||
"items = [1, 2, 3, 4]\n",
|
||
"k = 2\n",
|
||
"\n",
|
||
"print(\"my func\\n\")\n",
|
||
"\n",
|
||
"print(\"===Permutations===\")\n",
|
||
"measure(permutations, items)\n",
|
||
"print(\"===Combinations===\")\n",
|
||
"measure(combinations, items, k)\n",
|
||
"print(\"===Arrangements===\")\n",
|
||
"measure(arrangements, items, k)\n",
|
||
"\n",
|
||
"print(\"\\nitertools\\n\")\n",
|
||
"print(\"===Permutations===\")\n",
|
||
"measure(itertools.permutations, items)\n",
|
||
"print(\"===Combinations===\")\n",
|
||
"measure(itertools.combinations, items, k)\n",
|
||
"print(\"===Arrangements===\")\n",
|
||
"measure(itertools.permutations, items, k)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"id": "awwMLMoylUED"
|
||
},
|
||
"source": [
|
||
"**Задание 2**. В новую школу не успели завезти парты в один из классов. Поэтому в этот класс принесли круглые столы из столовой. Столы в столовой разных размеров — на 4, 7 и 13 человек, всего их хватало на 59 человек. Когда часть столов отнесли в класс, в столовой осталось 33 места. Какие столы могли быть отнесены в класс?"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 37,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"Столы по 4 места: 0 | Столы по 7 мест: 0 | Столы по 13 мест: 2\n",
|
||
"Столы по 4 места: 3 | Столы по 7 мест: 2 | Столы по 13 мест: 0\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"def find_possible_tables():\n",
|
||
" total_seats = 59\n",
|
||
" remaining_seats = 33\n",
|
||
" table_sizes = [4, 7, 13]\n",
|
||
"\n",
|
||
" for counts in itertools.product(range(total_seats // 4 + 1),\n",
|
||
" range(total_seats // 7 + 1),\n",
|
||
" range(total_seats // 13 + 1)):\n",
|
||
" if sum(count * size for count, size in zip(counts, table_sizes)) == total_seats-remaining_seats:\n",
|
||
" yield counts\n",
|
||
"\n",
|
||
"\n",
|
||
"for tables in find_possible_tables():\n",
|
||
" print(\"Столы по 4 места:\", tables[0], end=\" | \")\n",
|
||
" print(\"Столы по 7 мест:\", tables[1], end=\" | \")\n",
|
||
" print(\"Столы по 13 мест:\", tables[2])\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"**Задание 3**. Продавец имеет достаточное количество гирь для взвешивания следующих номиналов: 5гр, 10гр, 20гр, 50гр. каждый день к нему в магазин заходит житель соседнего дома и покупает ровно 500гр докторской колбасы. Продавец решил в течение месяца использовать различные наборы гирек для взвешивания. Сможет ли он выполнить задуманное?"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 51,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"Да\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"def find_possible_weights():\n",
|
||
" total = 500\n",
|
||
" weights = [5, 10, 20, 50]\n",
|
||
" \n",
|
||
" for counts in itertools.product(range(total // 5 + 1),\n",
|
||
" range(total // 10 + 1),\n",
|
||
" range(total // 20 + 1),\n",
|
||
" range(total // 50 + 1)):\n",
|
||
" if sum(count * weight for count, weight in zip(counts, weights)) == total:\n",
|
||
" yield counts\n",
|
||
"\n",
|
||
"print(\"Да\" if len(list(find_possible_weights())) >= 31 else \"Нет\")"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"**Задание 4**. Сколько можно найти различных семизначных чисел, сумма цифр которых равна ровно 50?"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": null,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"26418\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"def find_numbers() -> list[int]:\n",
|
||
" possible_combinations = itertools.product(range(10), repeat=7)\n",
|
||
" valid_numbers = []\n",
|
||
"\n",
|
||
" for combination in possible_combinations:\n",
|
||
" if combination[0] != 0:\n",
|
||
" if sum(combination) == 50:\n",
|
||
" number = int(\"\".join(map(str, combination)))\n",
|
||
" valid_numbers.append(number)\n",
|
||
"\n",
|
||
" return valid_numbers\n",
|
||
"\n",
|
||
"\n",
|
||
"print(len(find_numbers()))\n"
|
||
]
|
||
}
|
||
],
|
||
"metadata": {
|
||
"colab": {
|
||
"provenance": [],
|
||
"toc_visible": true
|
||
},
|
||
"kernelspec": {
|
||
"display_name": "Python 3 (ipykernel)",
|
||
"language": "python",
|
||
"name": "python3"
|
||
},
|
||
"language_info": {
|
||
"codemirror_mode": {
|
||
"name": "ipython",
|
||
"version": 3
|
||
},
|
||
"file_extension": ".py",
|
||
"mimetype": "text/x-python",
|
||
"name": "python",
|
||
"nbconvert_exporter": "python",
|
||
"pygments_lexer": "ipython3",
|
||
"version": "3.11.0"
|
||
}
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 1
|
||
}
|