Question

Let $X$ be a set having $3$ elements and $Y$ be a set having $4$ elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X,$ then the value of $(\beta -\alpha )$ is

Answer 1

Number of one-one functions $:X\to Y$ is
$n\left(X\right)=3=nn\left(Y\right)=4=m\Rightarrow \alpha =4\times 3\times 2=24$

Number of onto functions $:Y\to X$ is
$n\left(Y\right)=4=mn\left(X\right)=3=n\beta =n^m-{(}^n{C}_1(n-1{)}^m-{}^n{C}_2(n-2{)}^m+\dots \left)\right(\because m>n)\Rightarrow \beta =3^4-[{}^3{C}_1(2{)}^4-{}^3{C}_2(1{)}^4]\Rightarrow \beta =81-45=36$