The probability that Arun solves a certain problem is 2/3 and the probability that Bina solves it is 3/5, and they work independently. What is the probability that exactly one of them solves it?

Exactly one means Arun solves it while Bina does not, or the other way round: (2/3)(2/5) + (1/3)(3/5) = 4/15 + 3/15 = 7/15. The option 13/15 answers 'at least one of them solves it', 1 − (1/3)(2/5); 2/5 = (2/3)(3/5) is the probability that BOTH solve it; and 2/15 is the probability that neither does. As a check the three disjoint cases add up: 2/15 + 7/15 + 6/15 = 1.