**Theorem** $S_{n}$ has [[Order of Algebraic Structure|order]] $n!$ **Proof** Any injective function from $\{ 1,2,\dots,n \}$ to itself is surjective because if distinct elements get sent to distinct elements then the number of elements that get 'hit' must be $n$. Let's count the injections. There are $n$ choices for $f(1)$, $n-1$ choices for $f(2)$... Hence there are $n!$ injections