Coefficient in expansion - Discrete Math 
Let $\alpha$ be the coefficient of $x^2y^2z^2$ in $ (x^2+3y-2z)^5$. Show that $800\leq|\alpha|<1100$.

By my calculation, $\alpha=\binom {5}{2} \times (3^{2})\binom {3}{2} \times(-2)^2 \binom {1}{2}$
However $\binom {1}{2}=0$, so $\alpha$ should be $0$.
What am I doing wrong?

$$
\begin{align}
1^1\cdot3^2\cdot(-2)^2\cdot\overbrace{\binom{5}{2,2}}^{\substack{\text{trinomial}\\\text{coefficient}}}
&=1^1\cdot3^2\cdot(-2)^2\cdot\frac{5!}{2!2!1!}\\
&=1080
\end{align}
$$
