Question

How many odd integers are there between 101 and 999 have distinct digits?

Answer 1

Every number will have only 3 digits. In the ones place, there are 5 choices. For every choice of digit made in the ones place, there are 8 possible choices for the hundreds place (you can choose from 1-9, omitting whatever is used in the ones place), and there are 8 possible choices for the tens place (you can choose from 0-9, omitting whatever has already been used elsewhere).

So there are $7\cdot8\cdot8=320$ such numbers.