Explanation

(b)
$\ ^{18}C_{15} + 2\left( \ ^{18}C_{16} \right) + \ ^{17}C_{16} + 1 = \ ^{n}C_{3}$
$\Longrightarrow \ ^{18}C_{15} + \ ^{18}C_{16} + \ ^{18}C_{16} + \ ^{17}C_{16} +^{17}C_{17} = \ ^{n}C_{3}$
$\Longrightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^{19}C_{16} + \ ^{18}C_{16} + \ ^{18}C_{17} = \ ^{n}C_{3}$
$\Longrightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^{19}C_{16} + \ ^{19}C_{17} = \ ^{n}C_{3}$
$\Longrightarrow \ ^{20}C_{17} = \ ^{n}C_{3} \Longrightarrow \ ^{20}C_{3} = \ ^{n}C_{3} \Longrightarrow n = 20$