Number of Permutations

Theorem
Let $S$ be a set of $n$ elements.

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

Then the number of $r$-permutations of $S$ is:
 * ${}^r P_n = \dfrac {n!} {\paren {n - r}!}$

When $r = n$, this becomes:
 * ${}^n P_n = \dfrac {n!} {\paren {n - n}!} = n!$

Using the falling factorial symbol, this can also be expressed:
 * ${}^r P_n = n^{\underline r}$