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 function from Y to X, then the value of  $\frac{1}{5!}(\beta -\alpha )$ is _______

$n\left(X\right)=5;n\left(Y\right)=7$

$\alpha =$ Number of one-one function from set X to Y 

=Number of ways of selecting five images from seven images and then assigning to five preimages  $={}^7{C}_5\times 5!$
$\beta =$Number of on to function Y to X
=Number of ways in which all five images are associated with seven preimages.
=Number of ways in which seven different objects can be distributed among five persons if each gets at least one object
So, possible group sizes for distribution are (1,1,1,1,3) and (1,1,1,2,2) 

Hence,  $\beta =\frac{7!}{(1!)3!4!}\times 5!+\frac{7!}{{(1!)}^3{(2!)}^{2}3!2!}\times 5!$

$$\begin{gathered} \therefore \frac{\beta -\alpha }{5!}=\frac{7!}{3!4!}+\frac{7!}{{(2!)}^{3}3!}-{}^7{C}_5 \\ ={}^7{C}_3+3^7{C}_3-{}^7{C}_2=4\times 35-21=119 \end{gathered}$$