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  "body": "\n## Definition\n\nLet $x$ be an element of a commutative ring, for example a real number, and let $k$ be a natural number.\nThe falling factorial of $x$ of order $k$ is\n$$(x)_k = x (x - 1) (x - 2) \\cdots (x - k + 1) = \\prod_{i=0}^{k-1} (x - i).$$\nFor $k = 0$ the product is empty, and $(x)_0 = 1$.\n\n## Remarks\n\nOther notations are $x^{\\underline{k}}$ and $P(n, k)$.\nFor natural numbers $n \\ge k$, $(n)_k = n! / (n - k)!$. For $k \u003e n$, $(n)_k = 0$.\nFor natural numbers $n$ and $k$, $(n)_k$ is the [[number of injections|0X0N833FF8]] from a $k$-element set to an $n$-element set.\n\n## Explanations\n\n### L1 human\n\nThe falling factorial starts like the [[factorial|0WFCGVMRRC]] but stops after $k$ factors.\nFor example, $(5)_3 = 5 \\cdot 4 \\cdot 3 = 60$ is the number of ways to award gold, silver and bronze among 5 runners.\n",
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