The co-efficient of x-12 in the expansion of(x+(y/x3))20is

Question

The co-efficient of x-12 in the expansion of(x+(y/x3))20is (A)  20C8 (B)  20C8y8 (C)  20C12 (D)  20C12 y12

Tardigrade Admin · Accepted Answer

Suppose x-12 occurs is (r + 1)th term. We have Tr+1 = 20 Cr x20-r ((y/x3))r = 20Cr x20 -4r yr This term contains x-12 if 20-4r=-12 or r = 8 . ∴ The coefficient of x-12 is 20C8 y8

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

$^{20} C_{8}$

$^{20} C_{8} y^{8}$

$^{20} C_{12}$

$^{20} C_{12}^{y} 12$

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

Q. The co-efficient of x−12 in the expansion of(x+x3y​)20is

20C8​

20C8​y8

20C12​

20C12​y12