Question

Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ 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 $\frac{1}{5!}(\beta -\alpha )$ is

Answer 1

$n\left(X\right)=5$ and $n\left(Y\right)=7$
$\alpha \to \text{number of one-one function from}X\text{to}Y\alpha ={}^7{C}_5\times 5!$
$\beta \to \text{number of onto function from}Y\text{to}X$

For onto function, Co-domain and range of the function should be equal therefore 2 cases are possible:
Case $1:3$ elements of $Y$ are mapped to any one element of $X$ and remaining 4 elements of $X$ and $Y$ are mapped with each other.
Case $2:4$ elements of $Y$ are mapped to any two elements of $X$ and remaining 3 elements of $X$ and $Y$ are mapped with each other.
$\therefore \beta =[(\frac{7!}{3!.(1!{)}^4.4!}+\frac{7!}{(2!{)}^2.2!.(1!{)}^3.3!}]5!={}^7{C}_3.4.5!$
Hence, $\frac{\beta -\alpha }{5!}=\frac{({}^7{C}_3.4-{}^7{C}_5)5!}{5!}=119$