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In how many ways, a party of 5 men and 5 women be seated at a circular table, so that no two women are adjacent?
- a)720
- b)14400
- c)2880
- d)1440
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
In how many ways, a party of 5 men and 5 women be seated at a circular...
Lets first place the men (M). '*' here indicates the linker of round table
* M -M - M - M - M *
which is in (5-1)! ways
So we have to place the women in between the men which is on the 5 empty seats ( 4 -'s and 1 linker i.e * )
SO 5 women can sit on 5 seats in (5)! ways or
1st seat in 5 ways
2nd seat 4
3rd seat 3
4th seat 2
5th seat 1
i.e 5*4*3*2*1 ways
So the answer is 5! * 4! = 2880
Most Upvoted Answer
In how many ways, a party of 5 men and 5 women be seated at a circular...
Solution:
To solve this problem, we can use the principle of inclusion and exclusion.
Step 1: Total number of ways to seat 10 people at a circular table is (10-1)! = 9! = 362880.
Step 2: Let's consider the case where all the women are seated together. In this case, we can treat the group of women as a single entity and seat them first. There are 5! ways to arrange the women within the group, and 5! ways to arrange the men around the table. Therefore, there are 5! x 5! = 14400 ways to seat the group of women together.
Step 3: However, we have overcounted the cases where two particular women are seated together. There are 5 ways to choose one of the women, and once we fix her position, there are 4! ways to arrange the other women around her. Then, there are 5! ways to arrange the men around the table. Therefore, there are 5 x 4! x 5! = 2400 ways to seat the group of women together with two particular women seated together.
Step 4: We have also overcounted the cases where two pairs of women are seated together. There are 5C2 = 10 ways to choose two pairs of women, and once we fix their positions, there are 2! ways to arrange each pair within their group. Then, there are 5! ways to arrange the men around the table. Therefore, there are 10 x 2! x 2! x 5! = 4800 ways to seat the group of women together with two pairs of women seated together.
Step 5: However, we have subtracted too much in Step 4, because we have counted the case where all the women are seated together twice. Therefore, we need to add back the number of ways to seat all the women together. We already calculated this in Step 2, so we need to add back 14400.
Step 6: The total number of ways to seat the 10 people at a circular table without any two women seated together is:
9! - 14400 + 2400 - 4800 + 14400 = 2880.
Therefore, the correct answer is option C, 2880.
To solve this problem, we can use the principle of inclusion and exclusion.
Step 1: Total number of ways to seat 10 people at a circular table is (10-1)! = 9! = 362880.
Step 2: Let's consider the case where all the women are seated together. In this case, we can treat the group of women as a single entity and seat them first. There are 5! ways to arrange the women within the group, and 5! ways to arrange the men around the table. Therefore, there are 5! x 5! = 14400 ways to seat the group of women together.
Step 3: However, we have overcounted the cases where two particular women are seated together. There are 5 ways to choose one of the women, and once we fix her position, there are 4! ways to arrange the other women around her. Then, there are 5! ways to arrange the men around the table. Therefore, there are 5 x 4! x 5! = 2400 ways to seat the group of women together with two particular women seated together.
Step 4: We have also overcounted the cases where two pairs of women are seated together. There are 5C2 = 10 ways to choose two pairs of women, and once we fix their positions, there are 2! ways to arrange each pair within their group. Then, there are 5! ways to arrange the men around the table. Therefore, there are 10 x 2! x 2! x 5! = 4800 ways to seat the group of women together with two pairs of women seated together.
Step 5: However, we have subtracted too much in Step 4, because we have counted the case where all the women are seated together twice. Therefore, we need to add back the number of ways to seat all the women together. We already calculated this in Step 2, so we need to add back 14400.
Step 6: The total number of ways to seat the 10 people at a circular table without any two women seated together is:
9! - 14400 + 2400 - 4800 + 14400 = 2880.
Therefore, the correct answer is option C, 2880.
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In how many ways, a party of 5 men and 5 women be seated at a circular...
Total ways of men seated = 5-1! b'coz of circular arrangement now in the same way seating of women becomes 5! so total ways = 5!×4!=2880
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In how many ways, a party of 5 men and 5 women be seated at a circular table, so that no two women are adjacent?a)720b)14400c)2880d)1440Correct answer is option 'C'. Can you explain this answer?
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In how many ways, a party of 5 men and 5 women be seated at a circular table, so that no two women are adjacent?a)720b)14400c)2880d)1440Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In how many ways, a party of 5 men and 5 women be seated at a circular table, so that no two women are adjacent?a)720b)14400c)2880d)1440Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In how many ways, a party of 5 men and 5 women be seated at a circular table, so that no two women are adjacent?a)720b)14400c)2880d)1440Correct answer is option 'C'. Can you explain this answer?.
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