Difference between revisions of "Mock AIME 4 2006-2007 Problems/Problem 10"

Revision as of 10:30, 8 October 2007

Compute the remainder when

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

is divided by 1000.

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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	that sum equals <math>2^{2007}</math>. So we just need to find n such that		that sum equals <math>2^{2007}</math>. So we just need to find n such that

−	<math>n\equiv 2^{2007} \pmod ~~100~~</math>	+	<math>n\equiv 2^{2007} \pmod {1000}</math>