Problem:
A positive integer is called "ascending" if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there?
Solution:
An ascending positive integer must have distinct, nonzero digits. Thus the digits must be a subset of two or more elements from the set . Conversely, any subset of that has two or more elements corresponds to a unique ascending positive integer in which the elements of the subset are arranged in increasing order. It follows that the number of ascending positive integers is equal to the number of subsets of that have two or more elements. Since a nine-element set has subsets and ten of these subsets have fewer than two elements, the number of ascending positive integers is .
The problems on this page are the property of the MAA's American Mathematics Competitions