First slide

Binomial theorem for positive integral Index

Question

Coefficient of $x^{- 4} in {(\frac{3}{2} - \frac{3}{x^{2}})}^{10}$ is

Moderate

A

$\frac{405}{226}$
B

$\frac{504}{289}$
C

$\frac{450}{263}$
D

none of these

Solution

Suppose $x^{- 4}$ occurs in $(r + 1)^{th}$ term.
We have,
$1 T_{r + 1} =^{10} C_{r} {(\frac{3}{2})}^{10 - r} {(\frac{- 3}{x^{2}})}^{r} =^{10} C_{r} {(\frac{3}{2})}^{10 - r} (- 3)^{r} x^{- 2 r}$
This will contain $x^{4},$ if $- 2 r = - 4 \Rightarrow r = 2$
$∴$ Coefficient of $x^{- 4} =^{10} C_{2} {(\frac{3}{2})}^{10 - 2} (- 3)^{2} = \frac{3^{12} \times 5}{2^{8}}$

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