If $C_0,C_1,C_2,\dots ,C_n$ denote the binomial in the expansion of $(1+x{)}^n$ then $\sum _{r=0}^n \sum _{s=0}^n \left(C_r+C_s\right)=$ 

$\begin{array}{l}\sum _{r=0}^n \sum _{s=0}^n \left(C_r+C_s\right)\\ =\sum _{r=0}^n \sum _{s=0}^n C_r+\sum _{r=0}^n \sum _{s=0}^n C_s\end{array}$

$\begin{array}{l}=\sum _{s=0}^n \left(\sum _{r=0}^n C_r\right)+\sum _{r=0}^n \left(\sum _{s=0}^n C_s\right)\\ =\sum _{s=0}^n 2^n+\sum _{r=0}^n 2^n\\ =(n+1)2^n+(n+1)2^n=2(n+1)2^n=(n+1)2^{n+1}\end{array}$