Problem:
A committee of five is to be chosen from a group of 9 people. The probability that a certain married couple will either serve together or not at all is

a) 1/2
b) 5/9
c) 4/9
d) 2/3

Solution:

To find the probability that a certain married couple will either serve together or not at all, we need to consider two cases:

Case 1: The married couple serves together:

If the married couple serves together, they must both be included in the committee. We can choose the other three members from the remaining 7 people (excluding the married couple). The number of ways to select 3 members from a group of 7 is given by the combination formula: C(7, 3) = 7! / (3! * 4!) = 35.

Therefore, the number of ways to form a committee with the married couple serving together is 35.

Case 2: The married couple does not serve at all:

If the married couple does not serve at all, neither of them can be included in the committee. We can choose all 5 members from the remaining 7 people. The number of ways to select 5 members from a group of 7 is given by the combination formula: C(7, 5) = 7! / (5! * 2!) = 21.

Therefore, the number of ways to form a committee with the married couple not serving at all is 21.

Total number of possible committees:

The total number of possible committees that can be formed from a group of 9 people is given by the combination formula: C(9, 5) = 9! / (5! * 4!) = 126.

Probability:

The probability that a certain married couple will either serve together or not at all is given by the sum of the probabilities of the two cases divided by the total number of possible committees:

P = (Number of ways with the couple serving together + Number of ways with the couple not serving at all) / Total number of possible committees

P = (35 + 21) / 126 = 56 / 126 = 4 / 9

Therefore, the correct answer is option c) 4/9.