Properties of the Falling Factorial

In the setting of Finite Combinatorics: Notation, let $n,k\in\mathbb{N}_{0}$.

(i) Diagonal value.

$$(n)_{n}=n!.$$

(ii) Relation to binomial coefficients.

$$(n)_{k}=k!\binom{n}{k}.$$

(iii) Vanishing. If $k>n$, then $(n)_{k}=0$.

(iv) Positivity. If $k\le n$, then $(n)_{k}\ge1$.