Question

If, $C_0,C_1,C_2...........,C_n$ are the binomial coefficient in the expansion of $(1+x{)}^n,n$ being even, then $C_0+(C_0+C_1)+(C_0+C_1+C_2)+.......+(C_0+C_1+C_2+........+C_{n-1})$ is equal to

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

Answer 1

Answer: (B)

$n{.2}^{n-1}$

$C_0+(C_0+C_1)+(C_0+C_1+C_2)+....+(C_0+C_1+.....+C_{n-1})=S$ (let)

$S=n{C}_0+(n-1)C_1+(n-2)C_2+...+C_{n-1}$

$S=\frac{d}{dx}{(1+x)}^n$ at $x=1\Rightarrow n{(x+1)}^{n-1}=n.2^{n-1}$