Sum of coefficients of even powers of x in $(1+x)^5 (1+x^2)^5$? the exact question is asking the sum of coefficients of even powers of x in the expansion of $(1+x+x^2+x^3)^5$ and in the solution this expression is simplified to $(1+x)^5(1+x^2)^5$. The solution to this question says this expression is equal to $(1+5x+10x^2+10x^3+5x^4+x^5)(1+x^2)^5$, this I understood, then it says that required sum of coefficients is equal to $(1+10+5)2^5$. How did they get this?

The sum of coefficients of even power in $P(x)$ equals $$\frac{P(1)+P(-1)}{2}$$ in your case it's $$\frac{0+2^5\cdot 2^5}{2}=2^9=512$$.
