2010 Fermat Problem 25

Alex chose positive integers a, b, c, d, e, f and completely multiplied out the polynomial product

(1 - x)^a (1 + x)^b (1 - x + x^2)^c (1 + x^2)^d (1 + x + x^2)^e (1 + x + x^2 + x^3 + x^4)^f

After she simplified her result, she discarded any term involving x to any power larger than 6\ \text{and} was astonished to see that what was left was 1 - 2x. If a > d + e + f and b > c + d and e > c, what value of a did she choose?

\textbf{(A)}\ 17\quad \textbf{(B)}\ 19\quad \textbf{(C)}\ 20\quad \textbf{(D)}\ 21\quad \textbf{(E)}\ 23

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.