The coefficient of x^4 in the expansion of (1+x+x^2)^10 is ________.

Question

The coefficient of $x^4$ in the expansion of $(1+x+x^2)^{10}$ is \_\_\_\_\_\_\_\_.

Answer: 615

Accepted Answer

General multinomial term: $\dfrac{10!}{\alpha!\beta!\gamma!}x^{\beta+2\gamma}$, $\alpha+\beta+\gamma=10$. Need $\beta+2\gamma=4$:

- $\gamma=0,\beta=4,\alpha=6$: $\dfrac{10!}{6!4!0!}=210$
- $\gamma=1,\beta=2,\alpha=7$: $\dfrac{10!}{7!2!1!}=360$
- $\gamma=2,\beta=0,\alpha=8$: $\dfrac{10!}{8!0!2!}=45$

$$\text{Total}=210+360+45=615$$

Solutions · 1