A falling number is an integer whose decimal representation

A falling number is an integer whose decimal representation has the property that each digit except the units digit is larger than the one to its right.

For example 96521 is a falling number but 89642 is not.

How many n-digit falling numbers are there, for n= 5,6,and 7?

Solution

Each set of k digits can be arranged in exactly one way to form a falling number. Thus, there are just as many k-digit falling numbers as there are k-member subsets of { 5, 6, 7}. That is,

10C5 + 10C6 + 10C7 = 582