AMC 12 2006a Problem 24

The expression

$(x+y+z)^{2006}+(x-y-z)^{2006}$

is simplified by expanding it and combining like terms. How many terms are in the simplified expression?

$\mathrm{(A) \ } 6018 \mathrm{(B) \ } 671,676 \mathrm{(C) \ } 1,007,514 \mathrm{(D) \ } 1,008,016 \mathrm{(E) \ } 2,015,028$