Falling Factorial as Quotient of Factorials

Theorem
Let $x \in \Z_{\ge 0}$ be a positive integer.

Then:
 * $x^{\underline n} = \dfrac {x!} {\left({x - n}\right)!} = \dfrac {\Gamma \left({x + 1}\right)} {\Gamma \left({x - n + 1}\right)}$

where:
 * $x^{\underline n}$ denotes the $n$th falling factorial power of $x$.
 * $\Gamma \left({x}\right)$ denotes the Gamma function of $x$.

Also see

 * Rising Factorial as Quotient of Factorials