(a) In the expansion of ${(1 + x)^n}$, it is given that $^n{C_1}{,^n}{C_2}{,^n}{C_3}$ are in $A.P.$

==> ${2.^n}{C_2} = {\,^n}{C_1} + {\,^n}{C_3}$

==> $2.\frac{{n(n – 1)}}{{1.2}} = \frac{n}{1} + \frac{{n(n – 1)(n – 2)}}{{1.2.3}}$

==> $6(n – 1) = 6 + (n – 2)(n – 1)$

==> ${n^2} – 9n + 14 = 0$

==> $n = 2$or $n = 7$.

But $n = 2$ is not acceptable because, when $n=2$, there are only three terms in the expansion of ${(1 + x)^2}$, $\therefore $ $n = 7$.