AMC 12 2012a Problem 20

Consider the polynomial

$$P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)$$

The coefficient of $x^{2012}$ is equal to $2^a$. What is $a$?

\textbf{(A)}\ 5 \textbf{(B)}\ 6 \textbf{(C)}\ 7 \textbf{(D)}\ 10 \textbf{(E)}\ 24