we will consider two 10digit numbers to be equivalent if one

we will consider two 10-digit numbers to be equivalent if one can be obtained from the other by permuting its digits. how many nonequivalent 10-digit numbers are there?

Solution

If we consider two numbers to be equivalent is one can be obtained from the other by permuting its digits.
The way I went about it is I first figured there are 9,999,999,999 - 1,000,000,000 = 8,999,999,999 10-digit numbers out there (if leading 0\'s aren\'t allowed)...
But then I looked at it and thought that wouldn\'t it be true that only numbers made up of the same digits are nonequivalent, so there are 9 of those (1,111,111,111); and the numbers made up entirely of 0\'s except for the 1st digit, so there are 9 of those (2,000,000,000) (since we can\'t really move the non-zero digit anywhere or it wouldn\'t be a 10-digit number anymore).