Question

If $(1+x{)}^n=C_0+C_{1}x+\dots ..+C_n{x}^n,$ then the value of $\sum _{r=0}^n\sum _{s=0}^n(C_r+C_s)$ is equal to

• A. $(n+1)2^{n+1}$
• B. $(n-1)2^{n+1}$
• C. $(n+1)2^n$
• D. $(n-1)2^n$

Answer 1

The correct option is A $(n+1)2^{n+1}$

We have, $\sum _{r=0}^n\sum _{s=0}^n(C_r+C_s)$

$=\sum _{r=0}^n\sum _{s=0}^n{C}_r+\sum _{r=0}^n\sum _{s=0}^n{C}_s$

$=\sum _{s=0}^n\left(\sum _{r=0}^n{C}_r\right)+\sum _{r=0}^n\left(\sum _{s=0}^n{C}_s\right)$

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

$=(n+1)2^n+(n+1)2^n$
$=(n+1)2^{n+1}$