The coefficient of x^18 in the product (1+x)(1-x)^10(1+x+x^2)^9 is:

Question

The coefficient of $x^{18}$ in the product $(1+x)(1-x)^{10}(1+x+x^2)^9$ is:

Accepted Answer

$$(1-x)^{10}(1+x+x^2)^9(1+x)=(1-x)^9(1+x+x^2)^9(1-x)(1+x)=\left[(1-x)(1+x+x^2)\right]^9(1-x^2)$$

$=(1-x^3)^9(1-x^2)=(1-x^3)^9-x^2(1-x^3)^9$

Coefficient of $x^{18}$: from first term ($3k=18\Rightarrow k=6$): $\binom96$; from second term ($3k=16$, no integer solution): $0$.

$$\binom96=\frac{9!}{6!3!}=84$$

Solutions · 1