How many odd numbers between 1000 and 9999 have distinct digits?

For each of the 5 final digits that make the number odd (\(\{1, 3, 5, 7, 9\}\)) we can pick the first digit out of 8 (we exclude the final digit and 0, since the number cannot start with 0). The second and third digits can be picked from the remaining 8 digits (excluding the digit picked for the first position and the final digit, but including 0 this time) in \(8 \cdot 7 = 56\) ways, for the total of \(5 \cdot 8 \cdot 8 \cdot 7 = 2240\) numbers.