Finite permutation group/Cardinality/Fact/Proof

Let ${\displaystyle {}M=\{1,\ldots ,n\}}$. For ${\displaystyle {}1}$, there are ${\displaystyle {}n}$ possible images, for ${\displaystyle {}2}$, there are ${\displaystyle {}n-1}$ possible images remaining, for ${\displaystyle {}3}$, there are ${\displaystyle {}n-2}$ possible images remaining, etc. Therefore, there are altogether

${\displaystyle {}n(n-1)(n-2)\cdots 2\cdot 1=n!\,}$

possible permutations.