Find the coefficient of $x^m$ in the expansion $(1 + ax + bx^2)^n$

Question

Find the coefficient of $x^m$ in the expansion of $(1 + ax + bx^2)^n$

One approach would be:
Let $p = 1+ax$
and $q = bx^2$
Now expand $(p+q)^n$ and then expand $p$ and $q$ individually. But that is so clumsy. Can we derive a direct formula?

Answer 1

Hint: use the multinomial theorem: $$(a+b+c)^n=\sum_{i+j+k=n}\frac{n!}{i!j!k!}a^ib^jc^k$$