2006 AMC 12A 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\qquad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514\qquad \mathrm{(D) \ } 1,008,016\qquad \mathrm{(E) \ } 2,015,028

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.