Number of Permutations/Proof 2

Theorem

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

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


The number of $r$-permutations of $S$ is:

${}^n P_r = \dfrac {n!} {\paren {n - r}!}$


Using the falling factorial symbol, this can also be expressed:

${}^n P_r = n^{\underline r}$


Proof

By definition, an $r$-permutation of $S$ is an ordered selection of $r$ elements of $S$.

Also, an $r$-permutation of $S$ is a fortiori an injection from a subset of $S$ into $S$.


The result is immediate from Cardinality of Set of Injections‎.

$\blacksquare$

This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.