Let $\mathrm{A}=\left\{{\mathrm{x}}_1, {\mathrm{x}}_2, {\mathrm{x}}_3,.....,{\mathrm{x}}_7\right\},\mathrm{B}=\left\{{\mathrm{y}}_1, {\mathrm{y}}_2, {\mathrm{y}}_3\right\}$ . The total number of functions $\mathrm{f}:\mathrm{A}\to \mathrm{B}$ that are on to and there are exactly three elements ‘x’ in A such that $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{y}}_2$, is equal to

$\begin{array}{l}\mathrm{A}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4,{\mathrm{x}}_5,{\mathrm{x}}_6,{\mathrm{x}}_6,{\mathrm{x}}_7\right\},\mathrm{B}=\left\{{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{y}}_3\right\}\\ \mathrm{f}:\mathrm{A}\to \mathrm{B}\text{ is on to }\ni \mathrm{f}\left(\mathrm{x}\right)={\mathrm{y}}_2\end{array}$
Exactly 3 elements x in is  . This can be done in ${}^7\mathrm{C}_3$ways 
Remain A four elements in B₂ elements
$\therefore 2^4{-}^2{\mathrm{C}}_1(2-1{)}^4=14$
Total no. of on to functions ${=}^7{\mathrm{C}}_3\times 14$