2006 AMC 12A 第 24 题

A monomial $x^a y^b z^c$ survives only when $a$ is even; the odd-$a$ terms cancel

For each even $a$ from $0$ to $2006,$ count the possible values of $b$

A term $x^a y^b z^c$ survives only when $a$ is even, since terms with odd $a$ cancel between the two expansions.

For each even $a$ with $0 \le a \le 2006,$ the exponent $b$ ranges over $2007 - a$ values and $c = 2006 - a - b$ is then determined. Summing over even $a:$ $$\begin{gathered} (2007 - 0) + (2007 - 2) \\ {}+ \cdots + (2007 - 2006) \\ = 2007 + 2005 + \cdots + 1, \end{gathered}$$ the sum of the first $1004$ odd positive integers, which is $1004^2 = 1{,}008{,}016.$

Thus, the correct answer is D.