A group of 5 people are going to meet weekly at the library for 4 weeks. Each week, two people are selected at random to speak. Each person may speak in multiple weeks, but no pair of people will speak together more than once. The probability that there is a person who will never be asked to speak can be expressed as a/b where a and b are coprime positive integers. What is the value of a+b ?

To find the probability that there is a person who will never be asked to speak, let's consider each person individually.

In Week 1, there are five people and any two of them can be selected to speak. So, the first person has a chance of not being selected to speak of (5-1) out of 5.

In Week 2, there are four people remaining. We need to consider the case when the first person (who wasn't selected in Week 1) is chosen to speak in Week 2. The remaining three people can form two pairs, so the chance of the second person not being selected to speak is (3-1) out of 3.

In Week 3, there are three people remaining, and we need to consider the case when both the first and second people are selected to speak. The remaining person will not be asked to speak.

In Week 4, there are two people remaining, and we need to consider the case when the first two people are selected to speak. The remaining person will not be asked to speak.

To find the final probability, we multiply the probabilities together:

P(never asked to speak) = (4/5) * (2/3) * (1/2) * (1/2) = 8/60 = 2/15.

Thus, a = 2 and b = 15. The value of a + b is 2 + 15 = 17.