Difference between revisions of "Mock AIME 1 Pre 2005 Problems/Problem 1"

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Revision as of 12:00, 21 March 2008

Problem

Let $S$ denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when $S$ is divided by $1000$.

Solution

We find the sum of all possible hundreds digits, then tens digits, then units digits. Every one of $\{1,2,3,4,5,6,7,8,9\}$ may appear as the hundreds digit, and there are $9 \cdot 8 = 72$ choices for the tens and units digits. Thus the sum of the hundreds places is $(1+2+3+\cdots+9)(72) \times 100 = 45 \cdot 72 \cdot 100 = 324000$.

To finish later.

See also

Mock AIME 1 Pre 2005 (Problems, Source)
Preceded by
First question
Followed by
Problem 2
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