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  "body": "\n## Statement\n\nLet $X$ be a finite set with $|X| = n$ and let $k$ be a natural number.\nThen the number of subsets of $X$ with exactly $k$ elements is the [[binomial coefficient|092GEF38YD]] $\\binom{n}{k}$.\n\n## Remarks\n\nA proof by induction on $n$ fixes an element $x \\in X$. A $k$-subset either omits $x$ or contains $x$, and this gives [[Pascal's rule|0YRG241HQQ]].\nFor $k \u003e n$ both sides are $0$.\n\n## Explanations\n\n### L1 human\n\nTo pick a committee of $k$ people from $n$, first count the ordered lists of $k$ people. Then divide by the $k!$ orders of each committee.\nThis gives $\\frac{n (n-1) \\cdots (n-k+1)}{k!}$, which is $\\binom{n}{k}$.\n",
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