Counting Maps, Injections, Bijections and Permutations

In the setting of Finite Combinatorics: Notation, let $A$ and $B$ be finite sets with $\#A=k$ and $\#B=n$.

(i) Maps.

$$\#(B^{A})=n^{k}.$$

(ii) Injections.

$$\#\operatorname{Inj}(A,B)=(n)_{k}.$$

(iii) Bijections. If $k=n$, then $\#\operatorname{Bij}(A,B)=n!$; if $k\neq n$, then $\operatorname{Bij}(A,B)=\emptyset$.

(iv) Permutations.

$$\#S_{n}=n!.$$