Mulitnomial Coefficient Power on a number using the binomial coefficient $_nC_r=\binom nr$, find the coefficient of $(wxyz)^2$ in the expansion of $(2w-x+3y+z-2)^n$. The answer key says its $n=12$, $r= 2\times2\times2\times2\times4$ in one of the equation for $_nC_r$. Why is there a $4$ there ? is it because there are $4$ terms ?

The coefficient of $a^2b^2c^2d^2e^{n-8}$ in $(a+b+c+d+e)^n$ is the multinomial coefficient $\binom n{2,2,2,2,n-8}$. The $n-8$ is needed because the exponenents need to add up to $n$, anything else would make the multinomial coefficient undefined. For $n=12$ you get $n-8=4$, so I suppose that is where your $4$ comes from. Now put $a=2w,b=-x,c=3y,d=z,e=-2$ to get the real answer to your question.
