Question

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

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

Answer 1

The correct option is D $n\cdot 2^n$

We have,
$\sum _{r=0}^n\sum _{s=0}^n(C_r+C_s)=\sum _{r=0}^n(C_r+C_r)+2\underset{0\le r<s\le n}{\sum \sum }(C_r+C_s)$

$\Rightarrow (n+1)2^{n+1}=2\left(\sum _{r=0}^n{C}_r\right)+2\underset{0\le r<s\le n}{\sum \sum }(C_r+C_s)$

$\Rightarrow n\cdot 2^{n+1}=2\underset{0\le r<s\le n}{\sum \sum }(C_r+C_s)$

$\Rightarrow \underset{0\le r<s\le n}{\sum \sum }(C_r+C_s)=n\cdot 2^n$