Question

Find the coefficient of x^18 in the expansion of (1-x-x^2)^10

Answer 1

$(1-x-x^2)^{10}=(1-x-x^2)^{9}(1-x-x^2)=$

$=(1-x-x^2)\sum_{a+b+c=9, a,b,c\ge 0}1^a (-x)^b (-x^2)^c=(1-x-x^2) \times$

$\sum_{a+b+c=9, a,b,c\ge 0} (-1)^{b+c}x^{b+2c}=$

$=(x^{18}(-1)^9+x^{17}(-1)^9\cdot 9+x^{16}((-1)^8\cdot 9+(-1)^9\cdot \frac{9\cdot8}2)+\dots )(1-x-x^2)$

Hence, the answer is

$(-1)^9\cdot 1+(-1)^9\cdot9 \cdot(-1)+((-1)^8\cdot 9+(-1)^9\cdot \frac{9\cdot8}2)\cdot(-1)=35$