Question

Evaluate:$\sum _{m=0}^n\sum _{p=0}^m{}^n{C}_m{.}^m{C}_p$.

Answer 1

${\sum }_{r=0}^n{n}_{C_r}=2^n$ (Given)

So, ${\sum }_{m=0}^n{\sum }_{P=0}^m\left(n_{C_m}{m}_{C_P}\right)=n_{C_0}\left(0_{C_0}\right)+n_{C_1}(1_{C_0}+1_{C_1})+n_{C_2}(2_{C_0}+2_{C_1}+2_{C_2})+........+n_{C_n}(n_{C_0}+n_{C_1}+....+n_{C_n})$

Let $S={\sum }_{m=0}^n{\sum }_{P=0}^m\left(n_{C_m}{m}_{C_P}\right)=2^0{n}_{C_0}+2^1{n}_{C_1}+2^2{n}_{C_2}+......+2^n{n}_{C_n}$

$S=$ expansion of ${(1+x)}^n$ at $x=2$

So, $S=3^n$