Let $S=\{1,2,3,4,5....,2070\}$ and $A$ be the number of subsets of $S$ whose sum of elements in $S$ is divisible by $9$.If $$A=\frac{2^a\left(2^b+1\right)}{c}$$then find the values of $a,b$ and $c$.
The given solution is quite cryptic:
Let $f(x)=(1+x)(1+x^2)(1+x^3)....(1+x^{2070})$.
Then $A$ must be the sum of coefficient of $x^{9k}$ where $k$ is non-negative integer.So, $$A=\frac{2^{2070}+8.2^{230}}{9}$$How has the author of this question calculated this expression