$\text{ Let }A=\left\{x_1,x_2,x_3,x_4,x_5\right\}\text{ and }f:A\to A\text{ . The number of bijective functions such that }$

$f\left(x_i\right)=x_i\text{ for exactly two of the }{x}_i^{\mathrm{\prime }}s\text{ is }(i=1\text{ to }5)$

$\text{ Number of ways of two elements follow }\mathrm{f}\left(x_i\right)=x_i\text{ is }{5}_{{\mathrm{C}}_2}$

$\text{ Rest }3\text{ unit does not satisfy }\mathrm{f}\left(x_{\mathrm{i}}\right)={\mathrm{x}}_{\mathrm{i}}$

$\begin{array}{l}=\text{ Number of de-arrangement of }3\text{ things }\\ =3!\left(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}\right)=3!\left(\frac{1}{2}-\frac{1}{6}\right)=6\times \frac{3-1}{6}=2\end{array}$

$\text{ Required no of ways }=10\times 2=20$