Edited:

A correct solution is below by ps_dahiya...

Knowing odd+even =odd and odd+odd=even

Odd = {1,3,5,7,9}

Even= {2,4,6,8,0}

We have the following cases for odd sum of digits:

Case 1: 1 Odd & 5 Even

# of ways of picking 1 of 5 odd = 5

# of ways of picking 4 of 5 even = 5

Total = 25 numbers

Case 2: 3 Odd & 2 Even

# of ways of picking 3 of 5 odd = 5x4x3/3x2 = 10 ways

# of ways of picking 2 of 4 even = 4x3/2 = 6

Total number of numbers = 10x6 = 60.

Case 3: 5 Odd

# of ways of picking 5 of 5 =1

Adding all three cases= 5+1+60 = 66