Explanation

(b)
We have, ${(1 + x)}^{15} = C_{0} + C_{1}x + C_{2}x^{2} + ... + C_{15}x^{15}$
$\Rightarrow \frac{(1 + x)^{15} - 1}{x} = C_{1} + C_{2}x + ...C_{15}x^{14}$
On differentiating both sides w.r.t. $x$, we get
$\frac{x \bullet 15{(1 + x)}^{14} - {(1 + x)}^{15} + 1}{x^{2}} = C_{2} + 2C_{3}x + ... + 14C_{15}x^{13}$
On putting $x = 1$, we get
$C_{2} + 2C_{3} + ... + 14C_{15} = 15.2^{14} - 2^{15} + 1$
$= 13.2^{14} + 1$