If f is a function defined on the set of all non negative integers such that $f\left(0\right)=1,f\left(1\right)=0,f\left(n\right)+f(n-1)=nf(n-1)+(n-1)f(n-2)forn\ge 2thenf\left(5\right)=$

$\begin{array}{l}Givenf\left(0\right)=1,f\left(1\right)=0and\\ f\left(n\right)+f(n-1)=nf(n-1)+(n-1)f(n-2)\\ putn=2\\ f\left(2\right)+f\left(1\right)=2f\left(1\right)+f\left(0\right)\\ \Rightarrow f\left(2\right)+0=2\left(0\right)+1\Rightarrow f\left(2\right)=1\\ putn=3\\ f\left(3\right)+f\left(2\right)=3f\left(2\right)+2f\left(1\right)\\ f\left(3\right)+1=3+0\\ \Rightarrow f\left(3\right)=2\\ putn=4\end{array}$

$\begin{array}{l}f\left(4\right)+f\left(3\right)=4f\left(3\right)+3f\left(2\right)\\ \Rightarrow f\left(4\right)+2=8+3\\ \Rightarrow f\left(4\right)=9\\ putn=5\\ f\left(5\right)+f\left(4\right)=5f\left(4\right)+4f\left(3\right)\\ f\left(5\right)+9=45+8\\ f\left(5\right)=44\end{array}$