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 ?

Question

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 ?

Answer 1

To find the probability that there is a person who will never be asked to speak, we need to consider the number of ways people can be selected to speak over the 4 weeks.

First, let's find the total number of possibilities for selecting pairs of speakers each week. Since there are 5 people, we have 5 choose 2 (denoted as C(5,2)) which is equal to 10. This means there are 10 different pairs of speakers each week.

Now, let's analyze the possibilities for each person not being selected to speak. To calculate this, we need to subtract the cases where each person is selected at least once from the total number of possibilities.

For person A, there are C(8,2) possibilities where person A is not selected to speak with any other individual.

Similarly, for person B, there are C(8,2) possibilities where person B is not selected to speak with any other individual.

For person C, there are C(8,2) possibilities where person C is not selected to speak with any other individual.

For person D, there are C(8,2) possibilities where person D is not selected to speak with any other individual.

For person E, there are C(8,2) possibilities where person E is not selected to speak with any other individual.

Now, let's find the total number of possibilities where each person is selected at least once. We can obtain this value by subtracting the number of possibilities where at least one person is not selected from the total number of possibilities.

For a person not to be selected at all, there are C(8,2) possibilities for each of the 5 people. Hence, the total number of possibilities where at least one person is not selected is 5 * C(8,2).

Finally, the probability that there is a person who will never be asked to speak is given by:

(5 * C(8,2)) / (10 * 10 * 10 * 10)

To simplify this fraction, we calculate the individual values:

C(8,2) = 8! / (2! * (8-2)!) = 28

So the probability is:

(5 * 28) / (10^4) = 140 / 10000 = 7 / 500

Therefore, a = 7 and b = 500.

So the value of a + b is 7 + 500 = 507.