Integer to Power of Itself Falling is Factorial

Theorem

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

$n^{\underline n} = n!$

where:

$n^{\underline n}$ denotes the falling factorial
$n!$ denotes the factorial.

Proof

 $\displaystyle n^{\underline n}$ $=$ $\displaystyle \dfrac {n!} {\paren {n - n}!}$ Falling Factorial as Quotient of Factorials $\displaystyle$ $=$ $\displaystyle \dfrac {n!} {0!}$ $\displaystyle$ $=$ $\displaystyle n!$ Factorial of Zero

$\blacksquare$