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

Clearly, $\alpha {=}^7{C}_5\times 5!=2520$

  We knot that the number of onto functions from a set

 consisting of m elements to a set consisting of n elements $(m\ge n)$ 

is 

   $n^m{-}^n{C}_1(n-1{)}^m{+}^n{C}_2(n-2{)}^m{-}^n{C}_3(n-3{)}^m+\dots$ 

                                      $+{(-1{)}^{n-1}}^n{C}_{n-1}(n-(n-1){)}^n$

or  

            $\sum _{r=0}^n {(-1{)}^r}^n{C}_r(n-r{)}^m$

Here , $m=7$ and $n=5$ 

$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\beta =5^7{-}^5{C}_1\times 4^7{+}^5{C}_2\times 3^7{-}^5{C}_3\times 2^7{+}^5{C}_4\times 1^7=16800\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{\beta -\alpha }{5!}=\frac{16800-2520}{120}=119\end{array}$