2006 AMC 12A Problem 24

Question

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?

6018

671{,}676

1{,}007{,}514

1{,}008{,}016

2{,}015{,}028

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.

Problem 24 in other years