I've come across this problem and I cannot seem to find the answer. Let us call a number decreasing if its digits form a decreasing sequence (each digit is not larger than the preceding one). This means for example that 1, 22221111 and 888333300000 are decreasing numbers.
I know there exist infinitely many decreasing numbers as it does not many how many for example numbers 2 you take its digits still form a decreasing sequence and therefore a decreasing number. The problem I ran into is when we want to calculate how many decreasing numbers of ten digits there are.
I understand the total number of strictly decreasing numbers(so each digit is less than the preceding one) is equal to 1275 as that just is the sum of 10c1 , 10c2 up till 10c10. However I don't understand how to calculate the other number of possibilities we have to add to this number in order to account for the decreasing numbers.
I hope I explained my problem well enough, if not feel free to ask me to clearify some things.