Number of Terms in a Polynomial (4th Degree) Find the number of terms of $(x^3+5x^2-x+2)^4$, when like terms are added. 
My approach to this uses stars and bars to get $****|||$, since there are $4$ groups. 
$\binom{7}{3} = \frac{7\cdot6\cdot5}{6} = 35$
However, this is not the right answer.
What did I do wrong?

When you multiply and simplify you are going to get a sum of coefficients multipled by distinct powers of $x$, you just want to know what powers of $x$ are possible, the largest is $12 (x^3x^3x^3x^3)$ and the smallest is $0 (2\cdot2\cdot2\cdot2)$.
You just need to explain why all the ones in between $0$ and $12$ also happen.
