Question

If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+\dots +C_n{x}^n,$ then $\underset{0\le r<s\le n}{\sum \sum }(r+s)(C_r+C_s+C_r{C}_s)$ is equal to

• A. $n^2\cdot 2^n-\frac{n}{2}[2^{2n}-{}^{2n}{C}_n]$
• B. $n^2\cdot 2^n+\frac{n}{2}[2^{2n}-{}^{2n}{C}_n]$
• C. $n^2\cdot 2^n+\frac{n}{2}[2^{2n}+{}^{2n}{C}_n]$
• D. None of these

Answer 1

The correct option is B $n^2\cdot 2^n+\frac{n}{2}[2^{2n}-{}^{2n}{C}_n]$

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

$=n^2\cdot 2^n+\frac{n}{2}[2^{2n}-{}^{2n}{C}_n]$