Question

Find the coefficient of the term independent of x in ${(x-\frac{1}{x^2})}^{3n}$.

Answer 1

We can write the expression as $x^{3n}{(1-\frac{1}{x^3})}^{3n}$ which can be written as $x^{3n}\sum _{p=0}^{3n}$ ${}^{3n}{C}_r{\left(\frac{-1}{x^3}\right)}^{3n-p}$
Coefficient of $x^{3n+3p-9n}$ is $(-1{)}^{(n-p)3n}{C}_p$
To find coefficient of $x^{\theta }$, put $0=3n+3p-9n\Rightarrow p=2n$
$\therefore$ The coefficient of term independent of x is $(-1{)}^n$ ${}^{3n}{C}_{2n}$.