(a) Let the coefficient of three consecutive terms i.e. ${(r + 1)^{th}},\,\,\,\,{(r + 2)^{th}},\,\,\,{(r + 3)^{th}}$ in expansion of ${(1 + x)^n}$ are 165,330 and 462 respectively then, coefficient of ${(r + 1)^{th}}$ term $ = {}^n{C_r} = 165$

Coefficient of $(r + 2)^{th}$ term $ = {}^n{C_{r + 1}} = 330$ and

Coefficient of $(r + 3)^{th}$ term $ = {}^n{C_{r + 2}} = 462$
 $\frac{{{}^n{C_{r + 1}}}}{{{}^n{C_r}}} = \frac{{n – r}}{{r + 1}} = 2$

or $n – r = 2(r + 1)$ or $r = \frac{1}{3}(n – 2)$

and $\frac{{{}^n{C_{r + 2}}}}{{{}^n{C_{r + 1}}}} = \frac{{n – r – 1}}{{r + 2}} = \frac{{231}}{{165}}$

or $165(n – r – 1) = 231(r + 2)$ or $165n – 627 = 396r$

or $165n – 627 = 396 \times \frac{1}{3} \times (n – 2)$

or $165n – 627 = 132(n – 2)$ or $n = 11.$