Question

The value of the expression
$C_0^2-C_1^2+C_2^2-......+{(-1)}^n\times C_n^2$ is

• A. $0$, if $n$ is odd
• B. ${(-1)}^n$, if $n$ is odd
• C. ${(-1)}^{n/2}{{}^{n}C}_{n/2}$, if $n$ is even
• D. ${(-1)}^{n-1}{{}^{n}C}_{n-1}$, if $n$ is even

Answer 1

Answer: (A), (C)

$0$, if $n$ is odd
${(-1)}^{n/2}{{}^{n}C}_{n/2}$, if $n$ is even

Given , $C_0^2-C_1^2+C_2^2-......{(-1)}^n\times C_n^2$
Case I : When $n$ is odd, i.e. $n=2m+1$,
$S=C_0^2-C_1^2+C_2^2-.....+{(-1)}^{2m}\times C_{2m}^2+{(-1)}^{2m+1}\times C_{2m+1}$
$S=C_0^2-C_1^2+C_2^2-.....+C_{2m}^2-C_{2m+1}^2$
$=(C_0^2-C_{2m+1}^2)-(C_1^2-C_{2m}^2)+....+{(-1)}^m(C_m^2-C_{m+1}^2)$
But
${{}^{2m+1}C}_0={{}^{2m+1}C}_{2m+1};{{}^{2m+1}C}_1={{}^{2m+1}C}_{2m}$ etc.
Therefore, $S=0$
Case II: When $n$ is even, i.e. $n=2m$.
$S=C_0^2-C_1^2+C_2^2-......+{(-1)}^{2m}\times C_{2m}^2$
$S=C_0^2-C_1^2+C_2^2......+C_{2m}^2$
Consider the coefficient of the constant term in the following expression.
$[C_0-C_{1}x+C_2{x}^2-.....+{(-1)}^{2m}\times C_{2m}{x}^{2m}][C_0+C_1\frac{1}{x}+C_2\frac{1}{x^2}+....+C_{2m}\frac{1}{x^{2m}}]$
$={(1-x)}^{2m}{(1+\frac{1}{x})}^{2m}$
Coefficient of $x^{2m}$ in ${(1-x)}^{2m}{(1+x)}^{2m}$
$=$ Coefficient of $x^{2m}$ in ${(1-x^2)}^{2m}$
$={(-1)}^m{{}^{2m}C}_m={(-1)}^{n/2}{{}^{n}C}_{n/2}$