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?

Total number of sums is $101\choose2$=5050.  Subtract all ODD + EVEN combos: $50\times 51=2550$ confirms your answer of 2500
