How many pairs of two distinct integers chosen from the set {1, 2, 3, ..., 101} have a sum that is even? My Attempt:
N(even numbers between 1-101) = 50
N(odd numbers between 1-101) = 51
Since pair of two distinct integers that have a sum that is even can only be 
1. odd + odd
2. even + even

Thus, the answer is 51C2 + 50C2 = 2,500

Is my answer correct? If yes, how can I verify it?

Your answer is correct.  Good work.
To verify the answer, use the same method to solve a similar problem that you can check by hand.  For example:  

How many pairs of two distinct integers chosen from the set $\{1, 2, 3\}$ have a sum that is even?

Your same method says that the answer should be ${}_2C_2 + {}_1C_2 = 1$, and it is obvious that there is only one such pair, namely $1, 3$.
If the ${}_1C_2$ disturbs you, use $\{1,2,3,4\}$ instead or $\{1,2,3,4,5\}$; it still checks.
