Question

Find the coefficient of $x^{15}$ in ${(x^2+\frac{1}{x^3})}^{10}$

Answer 1

${(x^2+\frac{1}{x^3})}^{10}$

$\Rightarrow {}^{10}{C}_r(x^2{)}^{n-r}{\left(\frac{1}{x^3}\right)}^r$

$2\times 10-2r-3r=15$

$20-15=5r$

$r=1$

$\Rightarrow {}^{10}{C}_1(x{)}^9\times \left(\frac{1}{x^3}\right)$

$\Rightarrow \frac{10\times 9!}{9!\times 1}=10$