How many integers between 1000 and 9999 have distinct digits?

Distinct means different from all others. So there can be no repeated digits. Thus, 4124 is not possible because there are two '4' digits.

There are 9 ways to choose the first digit (1-9, as 0 is not possible). Subsequently, there are 9 choices for the second digit, since there are 10 possible digits (0-9), but we can't pick the same one as the first digit. Next, there are 8 ways to choose the 3rd digit, since we can't choose the same as either of the first two digits. Finally, there are 7 ways to choose the 4th digit.

The answer is 9 * 9 * 8 * 7 = 4536.