Rule for this type problem:

Choose the most restrictive thing(s) first!

Since the number must be odd, the last digit must
be odd.

So the last digit is the most restrictive thing to choose.

So we choose the last digit first:

We can choose the last, or 4th, digit 3 ways, 1, 5, or 9.
We can then choose the 1st digit 5 ways
(as any of the remaining 5 digits).
We can then choose the 2nd digit 4 ways
(as any of the remaining 4 digits).
We can then choose the 3rd digit 3 ways
(as any of the remaining 3 digits).

That's 3×5×4×3 = 180 ways.

Edwin