Question

The value of ${}^{2n}{C}_n{-}^n{C}_1{.}^{2n-2}{C}_n{+}^n{C}_2{.}^{2n-4}{C}_n-...$ is equal to

• A. $3^n$
• B. $4^n$
• C. $5^n$
• D. $2^n$

Answer 1

Answer: (D)

$2^n$

solve:

Let,

$S={}^{2n}Cn-{}^n{C}_1{}^{2n-2}{C}_n+{}^n{C}_2{}^{2n-4}{C}_n-\dots ..$

we know that ${}^n{c}_0=1$

$\Rightarrow S={}^n{C}_0{}^{2n}{C}_n-{}^n{C}_1{}^{2n-2}{C}_n+{}^n{C}_2{}^{2n-4}{c}_n-\dots$

$\Rightarrow S={}^n{C}_0[\mathrm{coff}\cdot$ of $x^n$ in $(1+x{)}^{2n}]-{}^n{C}_1[$ coff $\cdot$ of

$x^n\mathrm{in}(1+x{)}^{2n-2}]+\dots +{}^n{C}_n[\mathrm{coff}\cdot of{x}^{n}in(1+x{)}^{2n-2n}]$

$\Rightarrow S=$ Cofficient of $x^n$ in $\left[{}^n{C}_0\right(1+x{)}^{2n}-{}^n{C}_1(1+x{)}^{2n-2}+{}^n{C}_2(1+x{)}^{2n+4}+\dots ]$

$\Rightarrow S=\mathrm{coff}\cdot$ of $x^n$ in ${\left[\right(1+x{)}^2-1]}^n$

$\Rightarrow S=$ coff. of $x^n$ in ${[2x+x^2]}^n$

$\Rightarrow S=$ Coff. of $x^n$ in $x^n[x+2{]}^n$

So, coff. of $x^n$ in $x^n[x+2{]}^n$

$=2^n$