Question

How many four digit number can be formed such that the digits are in decreasing order?

Answer 1

Let's choose 4 distinct numbers from the set of digits {1,2,3,4,5,6,7,8,9}.{1,2,3,4,5,6,7,8,9}. (Note that 00 cannot be included in this set.) For each choice, there is a unique way by which we can relabel those 4 numbers so that they are in ascending order. (For example, if we chose the numbers 1,2,3, and 8,1,2,3, and 8, then the only way we can have them in ascending order would be 12381238 and nothing else.) So we'd have (94)(94) 4-digit numbers whose digits are strictly increasing. Using this logic, it wouldn't be too hard to extend it to any n-digit number as long as n<10.