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08 Jul 2025, 03:58
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Bunuel
There are n couples present and 2n people.
We need to form a committee of n people.
1. The number of different committees that can be formed if no two people who are married to each other are allowed to serve on the committee
we can select either person of the couple which can be done in 2 ways....the same is applicable for all (n) couples
Hence total number of different committees = 2*2*2*2.......n times = 2^n.
2. The number of different committees that can be formed if the committee must consist only of married couples.
Again we have to select only n people and couples have to be selected hence we need to choose n/2 couples out of n present which can be done in following way = nCn/2 = n!/ (n/2)![n-n/2]!
= n! / [ (n/2)! (n/2)!]
08 Jul 2025, 04:30
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Choose n=10
For no married couples it's possible to select one of the two people of the first couple, one of the two people of the second couple... one of the two people of the tenth couple.
2^10 -> answer is 2^n
For only married couples, there are 10 couples and the committee has 10 people, so 5 couples. In how many ways is possible to select 5 couples from a set of 10 couples?:
10C5=10!/(5!*5!) -> n!/((n/2)!*(n/2)!) = n!/((n/2)!)^2
The right answers are:
No married couples = 2^n
Only married couples = n!/((n/2)!)^2
For no married couples it's possible to select one of the two people of the first couple, one of the two people of the second couple... one of the two people of the tenth couple.
2^10 -> answer is 2^n
For only married couples, there are 10 couples and the committee has 10 people, so 5 couples. In how many ways is possible to select 5 couples from a set of 10 couples?:
10C5=10!/(5!*5!) -> n!/((n/2)!*(n/2)!) = n!/((n/2)!)^2
The right answers are:
No married couples = 2^n
Only married couples = n!/((n/2)!)^2
08 Jul 2025, 04:37
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a. No married couples allowed :
We need a total of n people total, so from every pair we can take exactly 1 person. From each pair, we have 2 choices. Doing this for n pairs gives 2^n possible committees. (option 3)
b. Only married couples allowed :
In this case, we must take both people or neither from any pair. To make a committee of n people, we therefore need n/2 full pairs.
Choosing n/2 pairs out of the n = n C (n/2)
-> n! / [(n/2)! * (n- (n/2))!]
-> n! / [(n/2!)^2] (Option 5)
We need a total of n people total, so from every pair we can take exactly 1 person. From each pair, we have 2 choices. Doing this for n pairs gives 2^n possible committees. (option 3)
b. Only married couples allowed :
In this case, we must take both people or neither from any pair. To make a committee of n people, we therefore need n/2 full pairs.
Choosing n/2 pairs out of the n = n C (n/2)
-> n! / [(n/2)! * (n- (n/2))!]
-> n! / [(n/2!)^2] (Option 5)
