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  "body": "\n## Statement\n\nLet $n$ and $k$ be natural numbers with $k \\le n$.\nThen\n$$\\binom{n}{k} = \\frac{n!}{k!\\,(n - k)!}.$$\n\n## Remarks\n\nEquivalently, $k!\\,(n-k)!\\,\\binom{n}{k} = n!$. In this form the identity has no division.\nThe formula needs $k \\le n$. For $k \u003e n$ the left side is $0$ and $(n - k)!$ has no meaning in the natural numbers.\nA cancelled form is $\\binom{n}{k} = \\frac{(n)_k}{k!}$, with the [[falling factorial|0N5M924N3P]] $(n)_k$.\n\n## Explanations\n\n### L1 human\n\nArrange all $n$ objects in a row in $n!$ ways, and take the first $k$ as the chosen set.\nEach choice of set appears $k!\\,(n-k)!$ times, because the first $k$ and the last $n-k$ objects can be reordered freely.\n",
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