Question

The coefficient of $x^{-10}$ in ${(x^2-\frac{1}{x^3})}^{10}$ is

• A. $-252$
• B. $210$
• C. $-\left(51\right)$
• D. $-120$

Answer 1

The correct option is B $210$

Let the term containing $x^{-10}$ in the expansion of

${(x^2-\frac{1}{x^3})}^{10}is{T}_{r+1}$
$\therefore T_{r+1}{=}^{10}{C}_r{\left(x^2\right)}^{10-r}{(-\frac{1}{x^3})}^r$

${=}^{10}{C}_r{x}^{20-5r}{(-1)}^r$

Since, this term contains $x^{-10}$

$\therefore 20-5r=-10$
$\Rightarrow 5r=30\Rightarrow r=6$
$\therefore$ Coefficient of $x^{-10}{=}^{10}{C}_6{(-1)}^6$
$=\frac{10!}{6!4!}$

$=210$

None option is correct.