96 lines
2.3 KiB
Markdown
96 lines
2.3 KiB
Markdown
---
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id: 5900f3a11000cf542c50feb4
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challengeType: 5
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title: 'Problem 53: Combinatoric selections'
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forumTopicId: 302164
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---
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## Description
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<section id='description'>
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There are exactly ten ways of selecting three from five, 12345:
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<div style='text-align: center;'>123, 124, 125, 134, 135, 145, 234, 235, 245, and 345</div>
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In combinatorics, we use the notation, $\displaystyle \binom 5 3 = 10$
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In general, $\displaystyle \binom n r = \dfrac{n!}{r!(n-r)!}$, where $r \le n$, $n! = n \times (n-1) \times ... \times 3 \times 2 \times 1$, and $0! = 1$.
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It is not until $n = 23$, that a value exceeds one-million: $\displaystyle \binom {23} {10} = 1144066$.
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How many, not necessarily distinct, values of $\displaystyle \binom n r$ for $1 \le n \le 100$, are greater than one-million?
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</section>
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## Instructions
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<section id='instructions'>
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</section>
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## Tests
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<section id='tests'>
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```yml
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tests:
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- text: <code>combinatoricSelections(1000)</code> should return a number.
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testString: assert(typeof combinatoricSelections(1000) === 'number');
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- text: <code>combinatoricSelections(1000)</code> should return 4626.
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testString: assert.strictEqual(combinatoricSelections(1000), 4626);
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- text: <code>combinatoricSelections(10000)</code> should return 4431.
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testString: assert.strictEqual(combinatoricSelections(10000), 4431);
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- text: <code>combinatoricSelections(100000)</code> should return 4255.
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testString: assert.strictEqual(combinatoricSelections(100000), 4255);
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- text: <code>combinatoricSelections(1000000)</code> should return 4075.
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testString: assert.strictEqual(combinatoricSelections(1000000), 4075);
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```
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</section>
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## Challenge Seed
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<section id='challengeSeed'>
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<div id='js-seed'>
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```js
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function combinatoricSelections(limit) {
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return 1;
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}
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combinatoricSelections(1000000);
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```
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</div>
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</section>
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## Solution
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<section id='solution'>
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```js
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function combinatoricSelections(limit) {
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const factorial = n =>
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Array.apply(null, { length: n })
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.map((_, i) => i + 1)
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.reduce((p, c) => p * c, 1);
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let result = 0;
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const nMax = 100;
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for (let n = 1; n <= nMax; n++) {
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for (let r = 0; r <= n; r++) {
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if (factorial(n) / (factorial(r) * factorial(n - r)) >= limit)
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result++;
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}
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}
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return result;
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}
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```
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</section>
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