Question

If $A{=}^{2n}{C}_0{.}^{2n}{C}_1{+}^{2n}{C}_1{.}^{2n-1}{C}_1{+}^{2n}{C}_2{.}^{2n-2}{C}_1+...,$ then $A$ is

• A. $0$
• B. $2^n$
• C. $n\cdot 2^{2n}$
• D. $1$

Answer 1

Answer: (C)

$n\cdot 2^{2n}$

$A=$ coefficients of $x$ in $[{}^{2n}{C}_0{(1+x)}^{2n}{+}^{2n}{C}_1{(1+x)}^{2n-1}+....]$

$=$ coefficients of $x$ in ${[1+(1+x)]}^{2n}$

$=$ coefficients of $x$ in ${(2+x)}^{2n}$

$=$ coefficients of $x$ in $2^{2n}{(1+\frac{x}{2})}^{2n}=n.2^{2n}$.