jmc

The Binomial Theorem says that, to find the coefficient of $x^a{y}^b$ in the expansion of $(x+y{)}^{a+b}$, we choose $a$ of the $(a+b)$ terms to contribute $x$ to the product, then the other $b$ terms contribute a $y$. Similarly, to find the coefficient of $x^3{y}^3{z}^2$ in $(x+y+z{)}^8$, we choose 3 of the 8 $(x+y+z)$ terms in the product $(x+y+z{)}^8$ to contribute an $x$ (which we can do in $\left(\genfrac{}{}{0ex}{}{8}{3}\right)$ ways), 3 of the remaining 5 terms to contribute a $y$ (which we can do in $\left(\genfrac{}{}{0ex}{}{5}{3}\right)$ ways), and the remaining 2 terms contribute $z$'s to the product. Thus, our answer is $$\left(\genfrac{}{}{0ex}{}{8}{3}\right)\left(\genfrac{}{}{0ex}{}{5}{3}\right)=56\cdot 10=\boxed{560}$$