Question

The co-efficient of $x^{-12}$ in the expansion of$\left(x+\frac{y}{x^3}\right)^{20}$is

• A. $\,^{20}C_8$
• B. $\,^{20}C_8y^8$
• C. $\,^{20}C_{12}$
• D. $\,^{20}C_{12}\,^y12$

Answer 1

Answer: (B)

$\,^{20}C_8y^8$

Answer 2

Suppose $x^{-12}$ occurs is (r + 1)th term. We have $T_{r+1} = ^{20} C_{r} x^{20-r} \left(\frac{y}{x^{3}}\right)^{r}$ $= ^{20}C_{r} x^{20 -4r} y^{r}$ This term contains $x^{-12}$ if $20-4r=-12$ or $r = 8$ . $\therefore$ The coefficient of $x^{-12}$ is ${^{20}C_{8}} \, y^{8}$