Mock AIME 3 Pre 2005 Problems/Problem 2

Problem

Let denote the number of digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when is divided by . (Repeated digits are allowed.)

Solution

Since the digits must be in increasing order, they must all be non-zero. We choose 7 digits out of 9, and when we do, they have only one order, so we choose them regardless of order, or .