a) well since there only four different types of exams and there are five applicants, then according to the pigeon hole principle, there has to be at least two people that take the same exam. Thus it is 100% likely that two of the five people take the same test. Or in another way of looking at this is by assigning each person a certain probability. Let's say that we bring this five people and order them in any way we'd like, and we bring the four tests and order them in any way we'd like( People: 1,2,3,4,5; Exams: A,B,C,D). And now let's see if it's possible that no person takes the same test, so we'll focus on just exam A and we'll like to give exam A to the first person, thus we'll say that the probability of person 1 getting exam A is 1/4. Next. we'll give person 2 any exam except exam A since person 1 already has exam A, thus the probability of person 2 getting one exam out of the leftover exams is 3/4. And now person 3 will have to get an exam that is different from person 1 and 2, thus they have a probability of 2/4 to get a unique exam. person 4 will have to get a unique exam as well, thus their probability is also 1/4. And finally person 5 has to get a unique exam, but since there are no more unique exams then person 5 has a probability of 0/4. Thus P(p1 and p2 and p3 and p4 and p5), assuming these are independent events, is 1/4 * 3/4 * 2/4 * 1/4 * 0/4 = 0. Thus there is no way that all five people can have different exams, and by contradiction, this means that at least two people must have the same exam.
b) in order to have both women take the same exam, we'll have to assume that they both take exam A. Thus, let's see what the probability is for none of the women to take the same exam and the men take any exam: 4/4 * 3/4 * 4/4 * 4/4 * 4/4. Now, since the probability of the two women taking the same exam is the compliment of the probability that we just took, we can say: P(both women taking same exam) = 1 - 3/4 = 1/4. So there is a 25% chance that both women take the same exam.
I hope this helps
