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\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24 $
1. We start with the polynomial given by:
   \[
   P(x) = \prod_{k=0}^{10}(x^{2^k} + 2^k) = (x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024).
   \]
   We need to find the coefficient of \(x^{2012}\) in this polynomial.

2. To find the coefficient of \(x^{2012}\), we need to consider the terms in the expansion of \(P(x)\) that will contribute to \(x^{2012}\). Each factor \((x^{2^k} + 2^k)\) can contribute either \(x^{2^k}\) or \(2^k\).

3. We need to select terms such that the exponents of \(x\) sum to 2012. We can write 2012 in binary to understand which powers of \(x\) we need:
   \[
   2012_{10} = 11111011100_2.
   \]
   This binary representation corresponds to:
   \[
   2012 = 2^{10} + 2^9 + 2^8 + 2^7 + 2^6 + 2^4 + 2^3 + 2^2.
   \]

4. We need to select \(x^{2^k}\) from the factors corresponding to \(k = 10, 9, 8, 7, 6, 4, 3, 2\) and the constant term \(2^k\) from the remaining factors \(k = 0, 1, 5\).

5. The coefficient of \(x^{2012}\) will be the product of the constants \(2^k\) from the factors where we did not select \(x^{2^k}\):
   \[
   2^0 \cdot 2^1 \cdot 2^5 = 1 \cdot 2 \cdot 32 = 64 = 2^6.
   \]

6. Therefore, the coefficient of \(x^{2012}\) is \(2^6\).

The final answer is \( \boxed{ a = 6 } \).