Explanation

(d)
Now, coefficient of $x^{15}$ in $(1 + x)^{20}$
$=$ coefficient of $x^{15}$ in $(1 + x)^{15}(1 + x)^{5}$
$\Longrightarrow \ \ ^{20}C_{15} =$coefficient of $x^{15}$ in $\left( \ ^{15}C_{0}x^{15} + \ ^{15}C_{1}x^{14} + \ ^{15}C_{2}x^{13} + \ ^{15}C_{3}x^{12} + \ ^{15}C_{4}x^{11} + \ ^{15}C_{5}x^{10} \right)$
${}^{}(^{5}C_{0}x^{5} + \ ^{5}C_{1}x^{4} + \ ^{5}C_{2}x^{3} + \ ^{5}C_{3}x^{2} + \ ^{5}C_{4}x^{\ } + \ ^{5}C_{5})$
$= = \ ^{20}C_{15} = \ ^{15}C_{0} \bullet \ ^{5}C_{5} + \ ^{15}C_{1} \bullet \ ^{5}C_{4} + \ ^{15}C_{2} \bullet \ ^{5}C_{3} + \ ^{15}C_{3} \bullet \ ^{5}C_{2} + \ ^{15}C_{4} \bullet \ ^{5}C_{1} + \ ^{15}C_{5} \bullet \ ^{5}C_{0}$
$\Longrightarrow \Longrightarrow \ ^{15}C_{0} \bullet \ ^{5}C_{5} + \ ^{15}C_{1} \bullet \ ^{5}C_{4} + \ ^{15}C_{2} \bullet \ ^{5}C_{3} + \ ^{15}C_{3} \bullet \ ^{5}C_{2} + \ ^{15}C_{4} \bullet \ ^{5}C_{1} = \ ^{20}C_{15} - \ ^{15}C_{5} \bullet \ ^{5}C_{0}$
$= = \frac{20!}{5!15!} - \frac{15!}{5!10!}$5_ ficient of terms