How many two-digit numbers have non-zero digits whose sum is odd? For example,54and38.

Two cases.....first digit  odd, second even

We have  5 choices for the  first  digit=  1,3,5,7,9

And we  have  4  choices for  the  second   2,4,6,8

Total  =  5 * 4   = 20

Second case......first digit  even,  second  odd

4 choices  for the  first digit  =2,4,6,8

5 choices for the  second  digit  = 1,3,5,7,9

So, again   ...20

So

20  +  20    =   40