Question

If in the expansion of ${\left(x^4-\frac{1}{x^3}\right)}^{15}$, $x^{-17}$ occurs in rth term, then
(a) r = 10
(b) r = 11
(c) r = 12
(d) r = 13

Answer 1

(c) r = 12

Here,
$$\begin{gathered} T_r={}^{15}C_{r-1}(x^4{)}^{15-r+1}{\left(\frac{-1}{x^3}\right)}^{r-1} \\ =(-1{)}^r\times {}^{15}C_{r-1}{x}^{64-4r-3r+3} \\ \mathrm{For}\mathrm{this}\mathrm{term}\mathrm{to}\mathrm{contain}{x}^{-17},\mathrm{we}\mathrm{must}\mathrm{have}: \\ 67-7r=-17 \\ \Rightarrow r=12 \end{gathered}$$