> [!NOTE] Theorem (Order of $S_{n}$) > Let $n\in \mathbb{N}^{+}.$ Let $S_{n}$ denote the [[Symmetric Groups of Finite Degree|nth symmetric group]]: that is, the set of all [[Permutation of Finite Degree|permutations of n letters]]. Then $S_{n}$ is [[Finite Group|finite]] with [[Cardinality|order]] $n!.$ **Proof**: Sufficient to count number of injections on $\mathbb{N}^{+}_{\leq n}$ since any injection from $\mathbb{N}^{+}_{\leq n}$ to itself is surjective (because if distinct elements get sent to distinct elements then the number of elements that get 'hit' must be $n$). There are $n$ choices for $f(1).$ Since $f(2)\neq f(1),$ there are $n-1$ choices for $f(2)$ once we've chosen $f(1).$ Continuing like this gives that the number of injections is $n\times(n-1)\times(n-2)\times\dots 1= n!.$ **Proof**: Follows from [[Number of k-Permutations of n Letters (Injections between Finite Sets)]].