How many four-digit positive integers have at least one digit that is a 2 or a 3?
I want to count how many four-digit numbers have at least one 2 or 3, but it might be easier to count how many do not have any 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 they range from 1000 to 9999. To count how many do not have any 2 or 3, I can think of the choices I have for each digit. For the first digit, I cannot choose 0, 2, or 3, so I have 7 choices. For the second digit, I cannot choose 2 or 3, so I have 8 choices. Similarly, for the third and fourth digits, I have 8 choices each. So the number of four-digit numbers that do not have any 2 or 3 is 7 * 8 * 8 * 8 = 3584. Therefore, the number of four-digit numbers that have at least one 2 or 3 is 9000 - 3584 = 5416. 