Functions

Number of Functions from $X$ to $Y$. $|X|=k$ $|Y|=n$

Number of all Functions

$|Y^X|=|Y{|}^{|X|}=n^k$

Number of Injective (One-to-One) Functions ($n\ge k$)

$n(n-1)(n-2)\cdots (n-k+1)=\frac{n!}{(n-k)!}$

Number of Surjective (Onto) Functions ($k\ge n$)

$n!\{\genfrac{}{}{0ex}{}{k}{n}\}={\sum }_{i=0}^{n-1}(-1{)}^i(\genfrac{}{}{0ex}{}{n}{i})(n-i{)}^k$

$\{\genfrac{}{}{0ex}{}{k}{n}\}=S(k,n)$ is Second kind (Stirling partition number)

Number of Bijective functions^[Theorem 3.30] ($k=n$)

$k!=n!$