The coefficient of $x^7$ in $(1-x+2x^3)^{10}$ is __________.
Step-by-Step Solution
Key Concept: Use multinomial theorem: $\frac{10!}{a!b!c!}(1)^a(-x)^b(2x^3)^c$ where $a+b+c=10$ and $b+3c=7$. Valid triples $(a,b,c)$: $(3,7,0),(5,4,1),(7,1,2)$.
Contributions: $\binom{10}{3,7,0}(-1)^7+\binom{10}{5,4,1}(-1)^4\cdot2+\binom{10}{7,1,2}(-1)^1\cdot4=-120+2520\cdot2\cdot\frac{1}{...}=960$.
Correct Answer: 960