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 (x^2-y+2z^3)^3

find the coefficient of x^2yz^3

so what I have done is

X=x^2

Y=-y

Z=2z^3

Meaning

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

y=-Y

thus

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

Thus the coefficient of XYZ should be 3!=6

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

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

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