Question

Let $A$ and $B$ be two sets such that $n\left(A\right)=6$ and $n\left(B\right)=3.$ If $x$ denotes the number of onto functions from $A$ to $B$ and $y$ denotes the number of one-one functions from $B$ to $A,$ then the value of $x-y$ is equal to

Answer 1

Given $n\left(A\right)=6,n\left(B\right)=3$

$x=$ Number of onto functions from $A$ to $B$
$={}^3{C}_0\cdot 3^6-{}^3{C}_1\cdot 2^6+{}^3{C}_2\cdot 1^6=540$
$y=$ Number of one-one functions from $B$ to $A$
$={}^6{P}_3=120$
$\therefore x-y=420$