If a + b is even, either both a and b are even, or both a and b are odd. We'll need to consider both possibilities.

Since a is chosen from between 1 and 5, the probability that a is even is 2/5, and the probability that it is odd is 3/5. Since b is chosen from between 6 and 10, the probability it is even is 3/5, and the probability that it is odd is 2/5.

Thus, the probability that both are even is (3/5)(2/5) = 6/25. The probability that both are odd is also (3/5)(2/5) = 6/25.

We don't care which of those things happens--we want one or the other to occur. Thus, we add the resulting probabilities: 6/25 + 6/25 = 12/25, choice (C).