What is the coefficient of $x^2y^2$ in the expansion of $(x+y)^4+(x+2y)^4$?
I know that the coefficient of $x^2y^2$ in the expansion of $(x+y)^4$ is given by the binomial formula: $\binom{4}{2}x^2y^2$. So the coefficient is $\binom{4}{2} = \frac{4!}{2!2!} = 6$. Similarly, the coefficient of $x^2y^2$ in the expansion of $(x+2y)^4$ is also given by the binomial formula: $\binom{4}{2}x^2(2y)^2$. So the coefficient is $\binom{4}{2} \times 2^2 = 24$. Therefore, the coefficient of $x^2y^2$ in the expansion of $(x+y)^4+(x+2y)^4$ is the sum of these two coefficients: $6+24 = 30$. 