Question

If $C_0,C_1,C_2\cdots ,C_n$ are Binomial Coefficients in ${(1+x)}^n$ , then value of the expression ${}^n{C}_0{}^n{C}_0{-}^{n+1}{C}_1{}^n{C}_1{+}^{n+2}{C}_2{}^n{C}_2+\cdots$ equals

• A. $-1$
• B. $0$
• C. ${(-1)}^n$
• D. $1$

Answer 1

The correct option is C ${(-1)}^n$

Let $n=2$ then we get

${}^2{C}_0.{}^2{C}_0-{}^3{C}_1{}^2{C}_1+{}^4{C}_2{}^2{C}_2$

$=1-(3\times 2)+\left(6\right)$

$=1$

$=(-1{)}^2$

Let $n=3$ then we get

${}^3{C}_0{}^3{C}_0-{}^4{C}_1.{}^3{C}_1+{}^5{C}_2.{}^3{C}_2-{}^6{C}_3.{}^3{C}_3$

$=1-12+30-20$

$=31-32$
$=-1$
$=(-1{)}^3$
Hence the correct answer is $(-1{)}^n$.