$\mathrm{Let}\mathrm{A}=\{1,2,3,4\}.\mathrm{The}\mathrm{number}\mathrm{of}\mathrm{functions}\mathrm{f}:\mathrm{A}\to \mathrm{A}\mathrm{satisfying}\mathrm{f}\left(\mathrm{f}\left(\mathrm{r}\right)\right)=\mathrm{r}\mathrm{for}\mathrm{all}1\le \mathrm{r}\le 4\mathrm{is}$

$\begin{array}{l}GivenA=\{1,2,3,4\}f:A\to Af\left(f\left(r\right)\right)=1\forall 1\le r\le 4\\ Iff\left(1\right)=1\Rightarrow f\left(f\left(1\right)\right)=f\left(1\right)=1\\ f\left(2\right)=1,2\Rightarrow f\left(2\right)=3,4\{\begin{array}{l}f\left(2\right)=3andf\left(3\right)=1\Rightarrow f\left(f\left(2\right)\right)=1\\ f\left(2\right)=4andf\left(4\right)=1\Rightarrow f\left(f\left(2\right)\right)=1\end{array}\\ Iff\left(3\right)\ne 3,1\Rightarrow f\left(3\right)=2,4\{\begin{array}{l}f\left(3\right)=2andf\left(2\right)=1\Rightarrow f\left(f\left(1\right)\right)=1\\ f\left(3\right)=4andf\left(4\right)=1\Rightarrow f\left(f\left(3\right)\right)=1\end{array}\\ Iff\left(4\right)=4,1\Rightarrow f\left(4\right)=2,3\{\begin{array}{l}f\left(4\right)=2\Rightarrow f\left(2\right)=1\Rightarrow f\left(f\left(4\right)\right)=1\\ f\left(4\right)=3\Rightarrow f\left(3\right)=1\Rightarrow f\left(f\left(4\right)\right)=1\end{array}\\ Iff\left(1\right)=2\Rightarrow f\left(2\right)=1andf\left(3\right)=f\left(4\right)=2\\ Iff\left(1\right)=3\Rightarrow f\left(3\right)=1andf\left(2\right)=f\left(4\right)=3\\ Iff\left(1\right)=4\Rightarrow f\left(4\right)=1andf\left(2\right)=f\left(3\right)=4\\ \therefore nooffunctions=10\end{array}$