I need help in calculating the sum of the coefficients of even powers of $x$ in $(1+x-2x^2)^6$

Question

I need to calculate the sum of coefficients of even powers of $x$ in $$(1+x-2x^2)^6$$

I don't know much about the multinomial theorem, but i know the basics pretty well. I have some ideas of solving such a question for a binomial expansion but that did not work well for multinomial expansions.

Answer 1

Let $$(1+x-2x^2)^6=a_0+a_1x+a_2x^2+\dots+a_{12}x^{12}$$ Substituting $x=1$, $$0=a_0+a_1+a_2+\dots+a_{12}$$ Substituting $x=-1$, $$32=a_0-a_1+a_2-\dots+a_{12}$$ Add the two to get your result.