Question

Find the coefficient of $x^{15}$ in the expansion of $(x-x^2{)}^{10}$

Answer 1

Let $x^{15}$ occur in the ${(r+1)}^{th}$ term. Then,

$\Rightarrow T_{r+1}={{}^{10}C}_r{x}^{10-r}{(-x^2)}^2$

$={{}^{10}C}_r{x}^{10-r}\times x^{2r}\times {(-1)}^r$

$={(-1)}^2\times {{}^{10}C}_r{x}^{10+r}$

$\therefore x^{10+r}=x^{15}$

$\therefore 10+r=15$ $\therefore r=15-10=5$

Required coefficient $={{}^{10}C}_5(-5)$

$=\frac{-10\times 9\times 8\times 7\times 6}{1\times 2\times 3\times 4\times 5}$

$=-252$

Hence, the answer is $-252.$