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Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place is
- a)10
- b)12
- c)14
- d)16
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Four couples (husband and wife) decide to form a committee of four mem...
The number of committees of 4 gentlemen =4C4 = 1
The number of committees of 3 gentlemen, 1 wife = 4C3 x 1C1
(∴ after selecting 3 gentlemen only 1 wife is left who can be included).
The number- of committees of 2 gentlemen, 2 wives
= 4C2 x 2C2
The number of committees of 1 gentlemen. 3 wives
= 4C1 x 3C3.
The number of committees of 4 wives = 1
∴ the required number of committes = 1 + 4 + 6 + 4+1 = 16
The number of committees of 3 gentlemen, 1 wife = 4C3 x 1C1
(∴ after selecting 3 gentlemen only 1 wife is left who can be included).
The number- of committees of 2 gentlemen, 2 wives
= 4C2 x 2C2
The number of committees of 1 gentlemen. 3 wives
= 4C1 x 3C3.
The number of committees of 4 wives = 1
∴ the required number of committes = 1 + 4 + 6 + 4+1 = 16
Most Upvoted Answer
Four couples (husband and wife) decide to form a committee of four mem...
Solution:
To form a committee of four members, we need to choose four individuals from the four couples. However, we need to ensure that no couple is selected together. Let's consider the problem step by step.
Step 1: Selecting the first member
We can select the first member in four ways, as there are four couples to choose from. Let's assume we choose couple A. Now, we have two options:
- Choose the husband from couple A
- Choose the wife from couple A
Step 2: Selecting the second member
After selecting the first member, we have three couples left to choose from. However, we need to make sure that we do not select the spouse of the first member. If we chose the husband from couple A in Step 1, we cannot select couple A again. Similarly, if we chose the wife from couple A in Step 1, we cannot select couple A again. Therefore, we have two options:
- Choose a couple other than couple A (if husband was chosen in Step 1)
- Choose a couple other than couple A (if wife was chosen in Step 1)
Step 3: Selecting the third member
After selecting the second member, we have two couples left to choose from. Again, we need to make sure that we do not select the spouse of the first and second members. Therefore, we have one option:
- Choose the couple that has not been chosen yet
Step 4: Selecting the fourth member
After selecting the third member, only one couple is left to choose from. Therefore, we have one option:
- Choose the remaining couple
Total number of committees:
To find the total number of committees, we multiply the number of options at each step.
Number of committees = (Number of options in Step 1) * (Number of options in Step 2) * (Number of options in Step 3) * (Number of options in Step 4)
Number of options in Step 1 = 4 (as we have four couples to choose from)
Number of options in Step 2 = 2 (as we have two couples remaining after selecting the first member)
Number of options in Step 3 = 1 (as only one couple is left to choose from)
Number of options in Step 4 = 1 (as only one couple is left to choose from)
Number of committees = 4 * 2 * 1 * 1 = 8
However, the question asks for the number of committees in which no couple finds a place. This means that if we selected couple A in Step 1, we cannot select couple A again in the subsequent steps. Therefore, we need to consider the cases where we chose the wife from couple A and the husband from couple A separately.
Number of committees if husband from couple A was chosen in Step 1 = 8
Number of committees if wife from couple A was chosen in Step 1 = 8
Total number of committees = Number of committees if husband was chosen + Number of committees if wife was chosen
Total number of committees = 8 + 8 = 16
Therefore, the correct answer is option D) 16.
To form a committee of four members, we need to choose four individuals from the four couples. However, we need to ensure that no couple is selected together. Let's consider the problem step by step.
Step 1: Selecting the first member
We can select the first member in four ways, as there are four couples to choose from. Let's assume we choose couple A. Now, we have two options:
- Choose the husband from couple A
- Choose the wife from couple A
Step 2: Selecting the second member
After selecting the first member, we have three couples left to choose from. However, we need to make sure that we do not select the spouse of the first member. If we chose the husband from couple A in Step 1, we cannot select couple A again. Similarly, if we chose the wife from couple A in Step 1, we cannot select couple A again. Therefore, we have two options:
- Choose a couple other than couple A (if husband was chosen in Step 1)
- Choose a couple other than couple A (if wife was chosen in Step 1)
Step 3: Selecting the third member
After selecting the second member, we have two couples left to choose from. Again, we need to make sure that we do not select the spouse of the first and second members. Therefore, we have one option:
- Choose the couple that has not been chosen yet
Step 4: Selecting the fourth member
After selecting the third member, only one couple is left to choose from. Therefore, we have one option:
- Choose the remaining couple
Total number of committees:
To find the total number of committees, we multiply the number of options at each step.
Number of committees = (Number of options in Step 1) * (Number of options in Step 2) * (Number of options in Step 3) * (Number of options in Step 4)
Number of options in Step 1 = 4 (as we have four couples to choose from)
Number of options in Step 2 = 2 (as we have two couples remaining after selecting the first member)
Number of options in Step 3 = 1 (as only one couple is left to choose from)
Number of options in Step 4 = 1 (as only one couple is left to choose from)
Number of committees = 4 * 2 * 1 * 1 = 8
However, the question asks for the number of committees in which no couple finds a place. This means that if we selected couple A in Step 1, we cannot select couple A again in the subsequent steps. Therefore, we need to consider the cases where we chose the wife from couple A and the husband from couple A separately.
Number of committees if husband from couple A was chosen in Step 1 = 8
Number of committees if wife from couple A was chosen in Step 1 = 8
Total number of committees = Number of committees if husband was chosen + Number of committees if wife was chosen
Total number of committees = 8 + 8 = 16
Therefore, the correct answer is option D) 16.
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Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place isa)10b)12c)14d)16Correct answer is option 'D'. Can you explain this answer?
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Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place isa)10b)12c)14d)16Correct answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place isa)10b)12c)14d)16Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place isa)10b)12c)14d)16Correct answer is option 'D'. Can you explain this answer?.
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