Explanation

(d)
We have $$$\sum_{r = 1}^{n}\ r^{2} \bullet \ ^{n\ }C_{r} = n(n - 1)2^{n - 2} + n \bullet 2^{n - 1}\ \ \ldots(i)$
and $\sum_{r = 1}^{n}\ {( - 1)}^{r - 1}r^{2}\ ^{n}C_{r} = 0$ …(ii)
On adding Eqs. (i) and (ii), we get
$2\left\lbrack 1\ ^{2}C_{1} + 3^{2}C_{3} + 5^{2}C_{5} + \ldots \right\rbrack = n(n - 1)2^{n - 2} + n \bullet 2^{n - 1}$
$\Longrightarrow 1^{2}C_{1} + 3^{2}C_{3} + 5^{2}C_{5} + ... = n(n - 1)2^{n - 3} + n \bullet 2^{n - 2}$