Explanation

(d)
$\ \ \ \ \ \ \ \ \ \ \ a_{1} = \ ^{n}C_{1},\ a_{2} = \ ^{n}C_{2}$
$\ \ \ \ \ \ \ \ \ \ \ a_{3} = \ ^{n}C_{3}$
∵ $a_{1},\ a_{2}and\ a_{3}\ are\ in\ AP$
$\Longrightarrow \ \ \ \ \ \ 2a_{2} = a_{1} + a_{3}$
$\Longrightarrow 2 \bullet \ ^{n}C_{2} = \ ^{n}C_{1} + \ ^{n}C_{3}$
$\Longrightarrow 2 \bullet \frac{n(n - 1)}{2!} = n + \frac{n(n - 1)(n - 2)}{3!}$
$\Longrightarrow \ n^{2} - 9n + 14 = 0$
$\Longrightarrow (n - 2)(n - 7) = 0$
$\Longrightarrow n = 7$ $(\because n = 2$is not Possible)