2006 AMC 12A Problem 24

Below is the professionally curated solution for Problem 24 of the 2006 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2006 AMC 12A solutions, or check the answer key.

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Concepts:binomial theorem parity basic counting

Difficulty rating: 2340

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?

$6018$

$671{,}676$

$1{,}007{,}514$

$1{,}008{,}016$

$2{,}015{,}028$

Solution:

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 $a:$ $(2007 - 0) + (2007 - 2) + \cdots + (2007 - 2006) = 2007 + 2005 + \cdots + 1,$ the sum of the first $1004$ odd positive integers, which is $1004^2 = 1{,}008{,}016.$

Thus, the correct answer is D.

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