Combinations for 4 letter word from the set {A,B,C,D} with at most one A, 2 B, 3C, and 4 Ds. No repetitions.

Question

Using synthetic multiplication of $(1+x)(1+x+x^2)(1+x+x^2)(1+x+x^2+x^3)(1+x+x^2+x^3+x^4)$, I looked for the coefficient of $x^4$ and got 15.

This is the correct way to go about it by generating functions right?