126 lines
2.6 KiB
Plaintext
126 lines
2.6 KiB
Plaintext
{
|
||
"cells": [
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Combinatoric selections (Euler Problem 53)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"collapsed": true
|
||
},
|
||
"source": [
|
||
"[https://projecteuler.net/problem=53](https://projecteuler.net/problem=53)\n",
|
||
"\n",
|
||
"There are exactly ten ways of selecting three from five, 12345:\n",
|
||
"\n",
|
||
"$123, 124, 125, 134, 135, 145, 234, 235, 245, 345$\n",
|
||
"\n",
|
||
"In combinatorics, we use the notation, 5C3 = 10.\n",
|
||
"\n",
|
||
"In general,\n",
|
||
"\n",
|
||
"$nCr = \\frac{n!}{r!(n−r)!}$, where r ≤ n, n! = n×(n−1)×...×3×2×1, and 0! = 1.\n",
|
||
"\n",
|
||
"It is not until n = 23, that a value exceeds one-million: 23C10 = 1144066.\n",
|
||
"\n",
|
||
"How many, not necessarily distinct, values of nCr, for 1 ≤ n ≤ 100, are greater than one-million?"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 3,
|
||
"metadata": {
|
||
"collapsed": true
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"import math\n",
|
||
"\n",
|
||
"def combinations_naiv(n, r):\n",
|
||
" return math.factorial(n) // (math.factorial(r) * math.factorial(n - r))\n",
|
||
"\n",
|
||
"assert(combinations_naiv(5, 3) == 10)\n",
|
||
"assert(combinations_naiv(23, 10) == 1144066)\n",
|
||
"assert(combinations_naiv(20, 20) == 1)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 7,
|
||
"metadata": {
|
||
"collapsed": true
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"count = 0\n",
|
||
"\n",
|
||
"for n in range (1, 101):\n",
|
||
" for r in range(1, n + 1):\n",
|
||
" if combinations_naiv(n, r) > 10**6:\n",
|
||
" count += 1\n",
|
||
"\n",
|
||
"s = count"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 9,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"4075\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"print(s)\n",
|
||
"assert(s == 4075)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": null,
|
||
"metadata": {
|
||
"collapsed": true
|
||
},
|
||
"outputs": [],
|
||
"source": []
|
||
}
|
||
],
|
||
"metadata": {
|
||
"completion_date": "Mon, 24 Dec 2018, 23:29",
|
||
"kernelspec": {
|
||
"display_name": "Python 3",
|
||
"language": "python3.6",
|
||
"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": "3.6.5"
|
||
},
|
||
"tags": [
|
||
"combinatoric",
|
||
"selection",
|
||
"factorial",
|
||
"airplane"
|
||
]
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 2
|
||
}
|