### 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: To count how many different 7-digit integers exist, I can think of filling up the 7 digits one by one, from left to right, and multiplying the number of choices I have at each step.
For the first digit, I have 9 choices, because it can't be 0.
It can be any of the digits from 1 to 9.
For the second digit, I have 10 choices, because it can be any of the digits from 0 to 9.
It doesn't matter what the first digit is.
Similarly, for the third, fourth, fifth, sixth, and seventh digits, I have 10 choices each.
So the total number of 7-digit integers is 9 times 10 times 10 times 10 times 10 times 10 times 10, which is 9 million.
Alternatively, I could have used the notation 10^6 to represent a million, and written the answer as 9 times 10^6.