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