### Instruction: How many different 7-digit positive integers exist? (Note that we don't allow "7-digit" integers that start with 0, such as 0123456; this is actually a 6-digit integer.).### Response: I need to count how many choices I have for each digit from left to right.
The first digit can't be 0, so I have 9 choices: 1, 2, 3, ..., 9.
The second digit can be any of the 10 digits: 0, 1, 2, ..., 9.
The same goes for the third, fourth, fifth, sixth, and seventh digits: 10 choices each.
To get the total number of possibilities, I need to multiply the number of choices for each digit: 9 x 10 x 10 x 10 x 10 x 10 x 10.