Question:
Find the coefficient of $x^{15}$ in the expansion of $\left(x-x^{2}\right)^{10}$.
Solution:
Given $\left(x-x^{2}\right)^{10}$
$T_{r+1}={ }^{10} \mathrm{C}_{r} x^{10-r}\left(-x^{2}\right)^{r}=(-1)^{r 10} \mathrm{C}_{r} x^{10-r} x^{2 r}=(-1)^{r 10} \mathrm{C}_{r} x^{10+r}$
For the coefficient of $x^{15}$, we have
$10+r=15 \Rightarrow r=5$
$T_{5+1}=(-1)^{510} C_{5} x^{15}$
Coefficient of $x^{15}=-\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 !}{5 \times 4 \times 3 \times 2 \times 1 \times 5 !}=-252$