Q: For the positive integers up to and including 2017, how many of those integers have at least one zero?
A: There are 
positive integers in total to consider.
No one digit numbers have a 0 (1-9).
Below is the list of the qualifying numbers:
| There are 9 two digit numbers: 10,20, 30…90 | 9 | |||||
| There are the numbers 100 – 110 and 200-210 up to 900 – 910 | 99 | |||||
| Also 120 130 140…190 and then 220 230 etc | 72 | |||||
| Then 1000-1099 | 100 | |||||
| Then same # of #s 1000 more than b | 99 | |||||
| Then same # of #s 1000 more than c | 72 | |||||
| 2000-2017 | 18 | |||||
| Total | 469 | |||||
Another solution is to exclude the numbers that do not have a zero and subtract from 2017:
9 one digit numbers
81 two digit numbers (9 x 9 : 9 choices for each digit 1-9)
729 three digit numbers (9 x 9 x 9: 9 choices for each digit 1-9)
For the numbers between 1000- 1999, there are the same amount as for the three digit numbers as they are the same numbers with a “1” in the thousands place value.
9 + 81 + 729 + 729 = 1548
2017 – 1548 = 469
Math Confidence by Robin the Math Lady 