Coefficient of x4 in the expansion of x2−3×210 is

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

A
$\frac{504}{256}$
B
$\frac{405}{256}$
C
$\frac{459}{512}$
D
$\frac{135}{256}$

Solution:

$T_{r + 1} =^{10} C_{r} {(\frac{x}{2})}^{10 - r} {(- \frac{3}{x^{2}})}^{r} =^{10} C_{r} \frac{(- 3)^{r}}{2^{10 - r}} x^{10 - 3 r}$

For the coefficient of $x^{4}$ set $10 - 3 r = 4$

$\Rightarrow r = 2 .$

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