If $C_0,C_1,C_2,\cdots C_n$ denotes the binomial coefficients in the expansion of $(1+x)^n$ then $\sum^n_{r=0}\sum^n_{s=0} (C_r+C_s)$ =?

Question

Problem :

If $C_0,C_1,C_2,\cdots C_n$ denotes the binomial coefficients in the expansion of $(1+x)^n$ then $\sum^n_{r=0}\sum^n_{s=0} (C_r+C_s)$ = ?

Solution :

We have : $\sum^n_{r=0}\sum^n_{s=0} (C_r+C_s)$

$= \sum^n_{r=0} \sum^n_{s=0} C_r +\sum^n_{r=0}\sum^n_{s=0}C_s$

$= \sum^n_{s=0}(\sum^n_{r=0}C_r) +\sum^n_{r=0}(\sum^n_{s=0}C_s)$

$= \sum^n_{s=0}2^n +\sum^n_{r=0}2^n$

$= n 2^n +n2^n $ $= 2n2^n$

But the answer is $(n+1)2^{n+1}$ please suggest where I am wrong, thanks.

Answer 1

$\sum_{r=0}^n \sum_{s=0}^n C_s = (n+1)2^n$. Then you have $2(n+1)2^n = (n+1)2^{n+1}$