SOLUTION: A committee of seven members is to be formed from a group of 11 members. The probability that a certain married couple will either serve together or not at all is.

Question 980630: A committee of seven members is to be formed from a group of 11 members. The probability that a certain married couple will either serve together or not at all is.
Found 2 solutions by Edwin McCravy, onlinepsa:
Answer by Edwin McCravy(20054) (Show Source): You can put this solution on YOUR website!
A committee of seven members is to be formed from a group of 11 members. The probability that a certain married couple will either serve together or neither serves.

The numerator is the number of ways they serve together or neither serves. Case 1. They serve together. Then we choose the remaining 5 to serve from the remaining 9 in 9C5 ways. Case 2. Neither serves. Then we choose the 7 to serve from the remaining 9 in 9C7 ways. Total successful ways: 9C5 + 9C7 = 126+36 = 162 Total ways to choose any 7: 11C7 = 330 Probability = 162/330 = 27/55 Edwin

Answer by onlinepsa(22) (Show Source): You can put this solution on YOUR website!
A committee of seven members is to be formed from a group of 11 members. The probability that a certain married couple will either serve together or not at all is.
.....
Method 1:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Married couple serving in the seven members committee:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Number of ways to do this= Number of ways to select the couple * Number of ways in which 5 more people are selected from remaining 9 members
= 1*9c5
= 126

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Married couple not serving in the seven members committee:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Consider selecting people to form the group of the people who are not forming the committee ( 4 members); if we are forming the non-committee members, rest of the members automatically form the committee

Number of ways to do this= Number of ways to select the couple in non-committee * Number of ways in which 2 more people are selected from remaining 11-2 or 9 members
= 1*9c2
= 26

Total probability= Total ways to select married couple in seven member committee or in the non-committee 'group' of 4 / Total ways to select 7 people out of 11
1.9c5 + 1.9c2
=------------------
11c7

162
=--------------------- = 81/165 = 27/55
330

Second Method:
----------------------

Here, we try to select any one from the couple to form the committee and find out the probability. The required answer will be 1 minus this probability.

Number of ways of chosing any one from the Married couple serving in the seven members committee: 2c1

Number of ways of chosing rest 6 members for the 7 member-committee: 9c6 [It is 9c6 and not 10c6 as once any one member from the couple is selected the other one cannot be put back into the group of remaining people (needs to be excluded)].

Probability to form the committee where any one from the couple is chosen = 2*9c6/11c7

Required probability =

1- 2*9c6/11c7
=1- 2*84/330
=1- 28/55
=27/55

Thanks,
PRD
https://onlinepsa.wordpress.com/
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