Question: Hey I would like to know the correct approach to this problem. I think I'm close but I'm confusing myself. 

How many sets of size five are there from the natural numbers 1-100 that contain exactly two odd numbers?

First we would select 2 odd numbers (50 choose 2)

Then sum it with (50 choose 3)? For the even #s?

Answer: 
Choose 2 odd numbers from 50 : $\binom{50}{2}$  

For each of these choices, you can choose 3 even numbers from the 50 even numbers : $\binom{50}{3}$ 

Thus it should be $\binom{50}{2}.\binom{50}{3}$
We want a set of size $5$, with $2$ odd numbers *and* $3$ even numbers. Converting this into math lingo, we get $\binom{50}{2} \cdot \binom{50}{3}$, and not their sum because we want both and not one or the other.