### Instruction: How many four-digit positive integers have at least one digit that is a 2 or a 3?### Response: I want to count how many four-digit numbers have at least one 2 or 3, but that might be hard to do directly.
Instead, I will try to count how many four-digit numbers have no 2 or 3, and then subtract that from the total number of four-digit numbers.
The total number of four-digit numbers is 9000, since the first digit can be from 1 to 9, and the other digits can be from 0 to 9.
To avoid having a 2 or 3, the first digit can be from 1, 4, 5, 6, 7, 8, or 9, which is 7 choices.
The other digits can be from 0, 1, 4, 5, 6, 7, 8, or 9, which is 8 choices.
So the number of four-digit numbers with no 2 or 3 is 7 times 8 times 8 times 8, which is 3584.
Therefore, the number of four-digit numbers with at least one 2 or 3 is 9000 minus 3584, which is 5416.