Binomial Theorem 1 Question 23
25. The coefficient of $x^{4}$ in $\frac{x}{2}-{\frac{3}{x^{2}}}^{10}$ is
(1983, 1M)
(a) $\frac{405}{256}$
(b) $\frac{504}{259}$
(c) $\frac{450}{263}$
(d) None of these
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Solution:
- The general term in $\frac{x}{2}-{\frac{3}{x^{2}}}^{10}$ is
$t _{r+1}=(-1)^{r}{ }^{10} C _r \frac{x}{2}^{10-r} \quad \frac{3}{x^{2}}{ }^{r}=(-1)^{r}{ }^{10} C _r \cdot \frac{3^{r}}{2^{10-r}} \cdot x^{10-3 r}$
For coefficient of $x^{4}$, we put $10-3 r=4$
$$ \begin{aligned} \Rightarrow & 3 r & =6 \\ \Rightarrow & r & =2 \end{aligned} $$
$\therefore$ Coefficient of $x^{4}$ in $\frac{x}{2}-\frac{3}{x^{2}}{ }^{10}=(-1)^{2} \cdot{ }^{10} C _2 \cdot \frac{3^{2}}{2^{8}}$
$$ =\frac{45 \times 9}{256}=\frac{405}{256} $$
