The coefficient of x^7 in (1-x+2x^3)^10 is _______.

Question

The coefficient of $x^7$ in $(1-x+2x^3)^{10}$ is \_\_\_\_\_\_\_.

Answer: 960

Accepted Answer

Multinomial general term: $\dfrac{10!}{a!b!c!}(-x)^b(2x^3)^c=\dfrac{10!}{a!b!c!}(-1)^b2^cx^{b+3c}$, with $a+b+c=10,\ b+3c=7$.

Solutions $(a,b,c)$: $(3,7,0),(5,4,1),(7,1,2)$.

$$\frac{10!}{3!7!0!}(-1)^7+\frac{10!}{5!4!1!}(-1)^42^1+\frac{10!}{7!1!2!}(-1)^12^2=-120+2520-1440=960$$

Solutions · 1