Rather than count the number of ways to put one or more 3's in a four-digit number, we can note that we can partition the set of all four-digit numbers into those which contain at least one 3, and those which contain no 3's.
Thus, the number of four-digit numbers which contain at least one 3 is the difference between the number of four-digit numbers and the number of four-digit numbers containing no 3's.
The total number of four-digit numbers can be calculated as the number of integers from 1 to 9999 less the total number of integers from 1 to 9999 with fewer than four digits:
"Total four-digit numbers" = 9999-999 = 9000
The total number of four-digit numbers with no 3's can be calculated by noting that there are 8 possible choices for the first digit: {1, 2, 4, 5, 6, 7, 8, 9} and 9 for the each of the remaining digits: {0, 1, 2, 4, 5, 6, 7, 8, 9}. Thus, multiplying, we get
"Total four-digit numbers with no 3's" = 8*9*9*9 = 5832
Finally, subtracting:
"Total four-digit numbers with">="one 3" = 9000-5832=3168
