Question

How many distinct 3 digit odd integers are there? Distinct meaning none of the digits repeat.

Answer 1

first digit: 9 choices (1-9)
then, 2nd digit: 9 choices (now 0 is allowed)
then, 3rd digit: 8 choices

so, 9*9*8

Answer 2

It has to be odd, so the 3rd digit has to be 1, 3, 5, 7, or 9. So there are only 5 options. Moreover, you do not consider the cases in which the digits do not repeat.

Answer 3

you got me on the odd value, but I do account for the non-repeatability of digits.

If they can be repeated, then there would be 9*10*10 possibilities for 3 digits.
My counting forces the preceding choices not to be used again.

Since the number must be odd, things get a bit more complicated, because we don't know whether the previous choices were odd or even.

You seem to have a handle on the problem. What is your reasoning?