Find the coefficient of

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 $x^{15}=-\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 !}{5 \times 4 \times 3 \times 2 \times 1 \times 5 !}=-252$

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