A three-digit number is odd if and only if its third digit is odd. 

Since the third digit is the most restrictive digit, we must pick
it first -- as any one of these three odd digits  {1, 3, 5}

So there are 3 ways to pick the third digit.  For each of those 3
ways to pick the third digit, there remain 5 ways to pick the 
first digit. So that's 3×5 ways to pick the third and first
digits.

Now for each of those 3×5 ways to pick the third amd first digit,
There remain 4 ways to pick the only remaining digit, the middle 
or second one. So that's 3×5×4 ways to pick all three digits.

Answer: 3×5×4 = 60 ways.

Edwin