Question

The term independent of $x$ in ${(x-\frac{1}{x^2})}^{3n}$ is

• A. $\frac{\left(3n\right)!}{{(n!)}^2}$
• B. $\frac{\left(3n\right)!{(-1)}^n}{{(2n!)}^2}$
• C. $\frac{\left(3n\right)!}{\left(n\right)!\left(2n\right)!}$
• D. $\frac{{(-1)}^n\left(3n\right)!}{\left(n\right)!\left(2n\right)!}$

Answer 1

The correct option is D $\frac{{(-1)}^n\left(3n\right)!}{\left(n\right)!\left(2n\right)!}$

Writing the general term, we get
$T_{r+1}=(-1{)}^r{}^{3n}{C}_r{x}^{3n-3r}$
For term independent of $x$
$3n-3r=0$
$\Rightarrow r=n$
Hence
The coefficient of the term independent of x will be
$(-1{)}^n{}^{3n}{C}_n$
$=(-1{)}^n\frac{3n!}{2n!.n!}$