## Multinomial Thereom

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?