Let $X$ be a D17: Finite set such that

(i)	$\text{Per}(X)$ is the D2921: Set of permutations on $X$

Let $\text{Inj}(X \to X)$ be the D2222: Set of injections from $X$ to itself. Since $X$ is finite, result R1855: Conditions for endomorphism on finite set to qualify as permutation shows that $\text{Per}(X) = \text{Inj}(X \to X)$. Therefore, applying R1854: Cardinality of the set of injections between finite sets yields $$\begin{equation} |\text{Per}(X)| = |\text{Inj}(X \to X)| = \frac{|X|!}{(|X| - |X|)!} = |X|! \end{equation}$$ $\square$