To find the number of odd numbers with distinct digits, I need to think about how to construct such a number. The last digit has to be odd, so it can be $1, 3, 5, 7, $ or $9$. The first digit can be any nonzero digit except the one I chose for the last digit, so there are $8$ choices for the first digit. The middle digit can be any digit except the ones I chose for the first and last digits, so there are $8$ choices for the middle digit as well. Therefore, the total number of odd numbers with distinct digits is $8 \times 8 \times 5 = 320$. 
How many odd numbers between $100$ and $999$ have distinct digits?