Question

The value of ${}^n{C}_0{}^n{C}_2{+}^n{C}_1{}^n{C}_3{+}^n{C}_2{}^n{C}_4+....+{}^n{C}_{n-2}{}^n{C}_n$ is equal to

• A. ${}^{2n}{C}_{n-2}$
• B. ${}^{2n}{C}_{n+1}$
• C. ${}^{2n}{C}_{n-1}$
• D. None of these

Answer 1

The correct option is A ${}^{2n}{C}_{n-2}$

${(1+x)}^n$.${(1+x)}^n$=(${}^n{C}_0$+${}^n{C}_1$x+${}^n{C}_2$.$x^2$................+${}^n{C}_{n-1}$.$x^{n-1}$+${}^n{C}_n$.$x^n$).(${}^n{C}_n$+${}^n{C}_{n-1}$x+${}^n{C}_{n-2}$.$x^2$................+${}^n{C}_2$.$x^{n-2}$+${}^n{C}_1$.$x^{n-1}$+${}^n{C}_0$.$x^n$)

Now if we multiply we get

(.............+$x^{n-2}$(${}^n{C}_0$.${}^n{C}_2$+${}^n{C}_1$.${}^n{C}_3$+..................+${}^n{C}_{n-2}$.${}^n{C}_n$...........................) $=$ coefficient of $x^{n-2}$ in expansion of ${(1+x)}^{2n}$ $=$ ${}^{2n}{C}_{n-2}$

nC0nC2+nC1nC3+nC2nC4+....+nCn−2nCn