Definition:Permutation

Mapping
A bijection $$f: S \to S$$ from a set $$S$$ to itself is called a permutation on (or of) $$S$$.

Ordered Selection
Let $$S$$ be a set of $$n$$ elements.

Let $$r \in \N: r \le n$$.

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 on a set of $$n$$ elements is $$\frac {n!} {\left({n-r}\right)!}$$.

It is often denoted $${}^m P_n$$ or $${}_m P_n$$.

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

Hence the number of $$r$$-permutations on a set of $$n$$ elements is $${}^n P_n = \frac {n!} {\left({n-n}\right)!} = n!$$.

Also see

 * Permutation on n Letters