Falling factorial

Definition

Let $x$ be an element of a commutative ring, for example a real number, and let $k$ be a natural number. The falling factorial of $x$ of order $k$ is $$(x)_k = x (x - 1) (x - 2) \cdots (x - k + 1) = \prod_{i=0}^{k-1} (x - i).$$ For $k = 0$ the product is empty, and $(x)_0 = 1$.

Remarks

Other notations are $x^{\underline{k}}$ and $P(n, k)$. For natural numbers $n \ge k$, $(n)_k = n! / (n - k)!$. For $k > n$, $(n)_k = 0$. For natural numbers $n$ and $k$, $(n)_k$ is the number of injections from a $k$-element set to an $n$-element set.

Explanations

L1 human

The falling factorial starts like the factorial but stops after $k$ factors. For example, $(5)_3 = 5 \cdot 4 \cdot 3 = 60$ is the number of ways to award gold, silver and bronze among 5 runners.