Question

Let $A$ and $B$ be two sets such that $n\left(A\right)=6$ and $n\left(B\right)=3,$ then number of onto functions from $A$ to $B$ is

Answer 1

If $n\left(A\right)=m$ and $n\left(B\right)=n,$ where $1\le n\le m,$ then number of onto functions from $A$ to $B$
$=\sum _{r=1}^n(-1{)}^{n-r}{}^n{C}_r\cdot r^m$

Here, $m=6,n=3$
$\therefore$ Number of onto functions
$=\sum _{r=1}^3(-1{)}^{3-r}{}^3{C}_r\cdot r^6$
$={}^3{C}_1-{}^3{C}_2\left(2^6\right)+{}^3{C}_3\left(3^6\right)$
$=3-192+729$
$=540$