First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

$\frac{405}{256}$
B

$\frac{504}{259}$
C

$\frac{450}{263}$
D

none of these

Solution

Suppose $x^{4}$ occurs in ${(r + 1)}^{t h}$ term.
we have ,
$T_{r + 1} =^{10} C_{r} {(\begin{matrix} x \\ 2 \end{matrix})}^{10 - r} {(- \frac{3}{x^{2}})}^{r} =^{10} C_{r} x^{10 - 3 r} (- 3)^{r} 2^{r - 10}$
This will contain $x^{4}$ , if
$∴ 10 - 3 r = 4 \Rightarrow 3 r = 6 \Rightarrow r = 2$
So, $x^{4}$ occurs in 3rd term and its coefficient is
$^{10} C_{2} \times (- 3)^{2} \times 2^{2 - 10} =^{10} C_{2} \times \frac{3^{2}}{2^{8}} = \frac{5 \times 9 \times 3^{2}}{2^{8}} = \frac{405}{256}$

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