Number of Permutations/Proof 2

Proof
From the definition, an $r$-permutation of $S$ is an ordered selection of $r$ elements of $S$.

It can be seen that an $r$-permutation is an injection from a subset of $S$ into $S$.

From Cardinality of Set of Injections‎, we see that the number of $r$-permutations ${}^n P_r$ on a set of $n$ elements is given by:
 * ${}^n P_r = \dfrac {n!} {\paren {n - r}!}$

From this definition, it can be seen that a bijection $f: S \to S$ is an $n$-permutation.

Hence the number of $n$-permutations on a set of $n$ elements is:
 * ${}^n P_r = \dfrac {n!} {\paren {n - n}!} = n!$