Mock AIME 4 2006-2007 Problems/Problem 10

Problem

Compute the remainder when

${2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}$

is divided by 1000.

Solution

Let $\omega$ and $\zeta$ be the two complex third-roots of 1. Then let

$S = (1 + \omega)^{2007} + (1 + \zeta)^{2007} + (1 + 1)^{2007} = \sum_{i = 0}^{2007} {2007 \choose i}(\omega^i + \zeta^i + 1)$ .

Now, if $i$ is a multiple of 3, $\omega^i + \zeta^i + 1 = 1 + 1 + 1 = 3$ . If $i$ is one more than a multiple of 3, $\omega^i + \zeta^i + 1 = \omega + \zeta + 1 = 0$ . If $i$ is two more than a multiple of 3, $\omega^i + \zeta^i + 1 = \omega^2 + \zeta^2 + 1= \zeta + \omega + 1 = 0$ . Thus

$S = \sum_{i = 0}^{669} 3 {2007 \choose 3i}$ , which is exactly three times our desired expression.

We also have an alternative method for calculating $S$ : we know that $\{\omega, \zeta\} = \{-\frac{1}{2} + \frac{\sqrt 3}{2}i, -\frac{1}{2} - \frac{\sqrt 3}{2}i\}$ , so $\{1 + \omega, 1 + \zeta\} = \{\frac{1}{2} + \frac{\sqrt 3}{2}i, \frac{1}{2} - \frac{\sqrt 3}{2}i\}$ . Note that these two numbers are both cube roots of -1, so $S = (1 + \omega)^{2007} + (1 + \zeta)^{2007} + (1 + 1)^{2007} = (-1)^{669} + (-1)^{669} + 2^{2007} = 2^{2007} - 2$ .

Thus, the problem is reduced to calculating $2^{2007} \pmod{1000}$ .

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Mock AIME 4 2006-2007 Problems/Problem 10

Problem

Solution