Mock AIME 4 2006-2007 Problems/Problem 10

Revision as of 11:30, 8 October 2007 by 1=2 (talk | contribs) (Solution)

Problem

Compute the remainder when

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

is divided by 1000.

Solution

Since


$\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{n}=2^n$


that sum equals $2^{2007}$. So we just need to find n such that

$n\equiv 2^{2007} \pmod {1000}$


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