The coefficient of x^10 in the expansion of (1+x^2-x^3^8 is

Choose $a$ factors contributing $x^2$ and $b$ factors contributing $-x^3$, the remaining $8-a-b$ contributing $1$. The power of $x$ is $2a+3b=10$, so $b$ must be even: $b=0$ gives $a=5$, and $b=2$ gives $a=2$ ($b=4$ would need $a<0$). The multinomial coefficients are $\frac{8!}{5!\,0!\,3!}=56$ for $(a,b)=(5,0)$, and $\frac{8!}{2!\,2!\,4!}\,(-1)^2=420$ for $(2,2)$. The coefficient is $56+420=476$, which agrees with a symbolic expansion.