To be odd, the last digit must be odd, that is, it must be 1,3, or 5.
To have three digits the first digit cannot be 0.

We choose the most restrictive digit first, which is the last one.

We can choose the last digit any of 3 ways.

Now we choose the next most restrictive digit, which is the first one.

That leaves 4 possible choices for the first digit, 2,4 and the two
odd digits besides the one chosen for the last digit.
So there are 3∙4=12 ways to choose the last and first digits.

The middle digit is the least restrictive one, so we choose it last.

So we finally choose the middle digit as any of the 4 digits besides the two
digits chosen for the first and last digits.
answer = 3∙4∙4 = 48 odd three-digit numbers. 

Here are all 48:

 1.  103
 2.  105
 3.  123
 4.  125
 5.  135
 6.  143
 7.  145
 8.  153
 9.  201
10.  203
11.  205
12.  213
13.  215
14.  231
15.  235
16.  241
17.  243
18.  245
19.  251
20.  253
21.  301
22.  305
23.  315
24.  321
25.  325
26.  341
27.  345
28.  351
29.  401
30.  403
31.  405
32.  413
33.  415
34.  421
35.  423
36.  425
37.  431
38.  435
39.  451
40.  453
41.  501
42.  503
43.  513
44.  521
45.  523
46.  531
47.  541
48.  543

Edwin