Question

If $(1+x{)}^n=C_0+C_{1}x+\cdots +C_n{x}^n,$ then the value of $\sum _{r=0}^n\sum _{s=0}^n{C}_r{C}_s$ is equal to

• A. $2^n$
• B. $2^{2n}$
• C. $2^{n+1}$
• D. $2^{3n}$

Answer 1

The correct option is B $2^{2n}$

$\sum _{r=0}^n\sum _{s=0}^n{C}_r{C}_s=\sum _{r=0}^n\left(C_r\sum _{s=0}^n{C}_s\right)=\left(\sum _{r=0}^n{C}_r\right)\left(\sum _{s=0}^n{C}_s\right)=2^n\cdot 2^n=2^{2n}$