Final answer:
There are 100 positive three-digit integers that contain the digit 9 at least once.
Step-by-step explanation:
The total number of three-digit positive integers is 900 (since the hundreds place can be filled with any digit from 1 to 9, the tens place can be filled with any digit from 0 to 9, and the ones place can also be filled with any digit from 0 to 9).
To count the number of three-digit integers that contain the digit 9 at least once, we need to subtract the number of three-digit integers that do not contain the digit 9 at all.
There are 8 possible digits that can be used in the hundreds place (1, 2, 3, 4, 5, 6, 7, and 8) and 10 possible digits that can be used in both the tens and ones place (0 to 8, and 0 to 9 respectively). So, the number of three-digit integers that do not contain the digit 9 is 8 * 10 * 10 = 800.
Therefore, the number of three-digit integers that contain the digit 9 at least once is 900 - 800 = 100.