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16 Dec 2024, 22:23
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Given the committee can include 2 married couples at most, let's look at the opposite possibility: the only configuration which is not suitable is 3 married couples.
How many ways are there to form a 6-people committee out of 3 pairs of spouses? Of course, only 1.
Hence, from the whole pool of possible arrangements we need to only subtract 1.
Overall, to get the committees, we need \(C^6_8 = \frac{8!}{6!2!}=\frac{56}{2}=28\) ways to explore.
Then, \(28-1=27\), and these are all the arrangements that are suitable for us.
Answer: D) 27
How many ways are there to form a 6-people committee out of 3 pairs of spouses? Of course, only 1.
Hence, from the whole pool of possible arrangements we need to only subtract 1.
Overall, to get the committees, we need \(C^6_8 = \frac{8!}{6!2!}=\frac{56}{2}=28\) ways to explore.
Then, \(28-1=27\), and these are all the arrangements that are suitable for us.
Answer: D) 27
16 Dec 2024, 22:25
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According to me
Total ways to select 6 members from a total of 8 people is 8C6 which will be 28
As the question stem restricts us to at most 2 married couples , so we will deduct the combination that will have all 3 married couples from the total no of combinations we calculated.
There fore, 8C6-3C3 --> 28-1
Ans: 27
Total ways to select 6 members from a total of 8 people is 8C6 which will be 28
As the question stem restricts us to at most 2 married couples , so we will deduct the combination that will have all 3 married couples from the total no of combinations we calculated.
There fore, 8C6-3C3 --> 28-1
Ans: 27
16 Dec 2024, 22:55
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3 married couples (MC) = 3x2= 6 people
2 bachelors (B)
Case 1: No married couples (0 couples)
Therefore, there are 6 different committees that can be formed with at most two married couples.
2 bachelors (B)
Case 1: No married couples (0 couples)
- We need to choose 6 members from the 2 bachelors.
- This can be done in C(2, 6) ways, which is 0 (since we can't choose 6 from 2).
- Choose 1 couple from 3: C(3, 1) = 3 ways.
- Choose 4 more members from the remaining 4 people (2 bachelors and 2 unmarried people from the other couples): C(4, 4) = 1 way.
- Total: 3 * 1 = 3 committees.
- Choose 2 couples from 3: C(3, 2) = 3 ways.
- Choose 2 more members from the remaining 2 people (2 bachelors): C(2, 2) = 1 way.
- Total: 3 * 1 = 3 committees.
Therefore, there are 6 different committees that can be formed with at most two married couples.
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From a group consisting of 3 married couples and 2 bachelors, a committee of 6 members is to be formed. How many different committees can be formed if the committee includes at most two married couples?
A. 12
B. 24
C. 25
D. 27
E. 28
From a group consisting of 3 married couples and 2 bachelors, a committee of 6 members is to be formed. How many different committees can be formed if the committee includes at most two married couples?
A. 12
B. 24
C. 25
D. 27
E. 28
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