Mock AIME 1 Pre 2005 Problems/Problem 1
Let denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when is divided by .
We find the sum of all possible hundreds digits, then tens digits, then units digits. Every one of may appear as the hundreds digit, and there are choices for the tens and units digits. Thus the sum of the hundreds places is .
Every one of may appear as the tens digit; however, since does not contribute to this sum, we can ignore it. Then there are choices left for the hundreds digit, and choices afterwards for the units digit (since the units digit may also be ). Thus, the the sum of the tens digit gives .
The same argument applies to the units digit, and the sum of them is . Then .
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