Explanation

(a)
We know that,
${(1 + x)}^{n} = C_{0} + C_{1}x + C_{2}x^{2} + ... + C_{n}x^{n}$
${(x - 1)}^{n} = C_{0}x^{n} - C_{1}x^{n - 1} + C_{2}x^{n - 2} - ... + {( - 1)}^{n}C_{n}$
On multiplying both equations and equating the coefficient of $x^{n}$, we get
$C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - ... + {( - 1)}^{n}C_{n}^{2} = \ ^{n}C_{\frac{n}{2}}{( - 1)}^{\frac{n}{2}}{(x^{2})}^{\frac{n}{2}}$
Above is possible only when $\frac{n}{2}$ is an integer $ie$, $n$ is even and in case $n$ is odd, then term $x^{n}$ will not occur