How to solve multiplication of $3$ generating functions?

Question

I'm trying to calculate the solution to this question. We can convert the original two-variable generating function into this format: $$ f(x)=(1+x+x^2)^{10}(1+x^3+x^6+\dots)^5 $$ which according to power series identities becomes: $$ \bigg(\frac{1-x^3}{1-x}\bigg)^{10}\bigg(\frac{1}{1-x^3}\bigg)^5=\\ \sum_{k=0}^{10}{10\choose k}(-1)^kx^{3k}\cdot \sum_{i=0}^{\infty} {10+i-1\choose i}x^i\cdot \sum_{j=0}^{\infty}{5+j-1\choose j}x^{3j} $$ In order to find the coefficient $x^7$, there're $6$ possible combinations of $x^{3k}x^ix^{3j}$ such that $3k+i+3j=7$. This seems quite complicated I'm wondering if I'm on the right track with the solution.