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  "body": "\n## Definition\n\nLet $n$ and $k$ be natural numbers.\nThe binomial coefficient $\\binom{n}{k}$, read \"$n$ choose $k$\", is the natural number defined by the recursion\n$$\\binom{n}{0} = 1, \\qquad \\binom{0}{k+1} = 0, \\qquad \\binom{n+1}{k+1} = \\binom{n}{k} + \\binom{n}{k+1}.$$\n\n## Remarks\n\nThe value $\\binom{n}{k}$ is the [[number of k-element subsets|0BFMR2FQ8B]] of an $n$-element set. Many texts take this as the definition.\nFor $k \u003e n$, $\\binom{n}{k} = 0$. For $k \\le n$, the [[factorial formula|033KTKKRV6]] gives $\\binom{n}{k} = \\frac{n!}{k!\\,(n-k)!}$.\nThe recursion is [[Pascal's rule|0YRG241HQQ]], and the values form [[Pascal's triangle|05DME4TZ3K]].\n\n## Explanations\n\n### L1 human\n\nThe number $\\binom{n}{k}$ tells how many ways there are to choose $k$ objects out of $n$ when the order does not matter.\nFor example, $\\binom{4}{2} = 6$: from $\\{1, 2, 3, 4\\}$ you can choose $12, 13, 14, 23, 24, 34$.\n",
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