119 lines
2.5 KiB
Plaintext
119 lines
2.5 KiB
Plaintext
{
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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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"source": [
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"# Combinatoric selections (Euler Problem 53)"
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]
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},
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{
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"cell_type": "markdown",
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"source": [
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"[https://projecteuler.net/problem=53](https://projecteuler.net/problem=53)\n",
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"\n",
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"There are exactly ten ways of selecting three from five, 12345:\n",
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"\n",
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"$123, 124, 125, 134, 135, 145, 234, 235, 245, 345$\n",
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"\n",
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"In combinatorics, we use the notation, 5C3 = 10.\n",
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"\n",
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"In general,\n",
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"\n",
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"$nCr = \\frac{n!}{r!(n−r)!}$, where r ≤ n, n! = n×(n−1)×...×3×2×1, and 0! = 1.\n",
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"\n",
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"It is not until n = 23, that a value exceeds one-million: 23C10 = 1144066.\n",
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"\n",
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"How many, not necessarily distinct, values of nCr, for 1 ≤ n ≤ 100, are greater than one-million?"
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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": 3,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"import math\n",
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"\n",
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"def combinations_naiv(n, r):\n",
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" return math.factorial(n) // (math.factorial(r) * math.factorial(n - r))\n",
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"\n",
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"assert(combinations_naiv(5, 3) == 10)\n",
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"assert(combinations_naiv(23, 10) == 1144066)\n",
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"assert(combinations_naiv(20, 20) == 1)"
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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": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"count = 0\n",
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"\n",
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"for n in range (1, 101):\n",
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" for r in range(1, n + 1):\n",
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" if combinations_naiv(n, r) > 10**6:\n",
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" count += 1\n",
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"\n",
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"s = count"
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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": 9,
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"metadata": {},
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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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"4075\n"
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]
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}
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],
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"source": [
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"print(s)\n",
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"assert(s == 4075)"
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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": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"completion_date": "",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python3.6",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.6.5"
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},
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"tags": []
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},
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"nbformat": 4,
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}
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