There is this question asking for a coeffecient of a certain expansion, however I dont seem to be getting the correct answer.

With the expansion of $\displaystyle (x^2-y+2z^3)^3$

find the coefficient of $\displaystyle x^2yz^3$

so what I have done is

$\displaystyle X=x^2$

$\displaystyle Y=-y$

$\displaystyle Z=2z^3$


$\displaystyle z^3=\frac{Z}{2}$

$\displaystyle y=-Y$


$\displaystyle x^2yz^3=X(-Y)(\frac{Z}{2})=-\frac{1}{2}(X)(Y)(Z)$

Thus the coefficient of $\displaystyle XYZ$ should be $\displaystyle 3!=6$

and then it should be multiplied by $\displaystyle -\frac{1}{2}$

thus $\displaystyle -\frac{6}{2}=-3$ <- which I think should be the binomial coefficient of $\displaystyle x^2yz^3$

However, i learned that the answer is -12
could anyone tell me where I have went wrong?