Question
The co-efficient of $${x^4}$$ in $${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$$ is
A.
$$\frac{{405}}{{256}}$$
B.
$$\frac{{504}}{{259}}$$
C.
$$\frac{{450}}{{263}}$$
D.
none of these
Answer :
$$\frac{{405}}{{256}}$$
Solution :
General term in the expansion $${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$$ is
$${T_{r + 1}} = {\,^{10}}{C_r}{\left( {\frac{x}{2}} \right)^{10 - r}}{\left( {\frac{{ - 3}}{{{x^2}}}} \right)^r} = {\,^{10}}{C_r}{x^{10 - 3r}}\frac{{{{\left( { - 1} \right)}^r}{3^r}}}{{{2^{10 - r}}}}$$
For co - eff of $${x^4},$$ we should have
$$\eqalign{
& 10 - 3r = 4 \cr
& \Rightarrow \,\,r = 2 \cr} $$
∴ Co - eff of $${x^4} = {\,^{10}}{C_2}\frac{{{{\left( { - 1} \right)}^2}{3^2}}}{{{2^8}}} = \frac{{405}}{{256}}$$