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The number of ways of arranging 8 men and 4 women around a circular table such that no two women can sit together, is
- a)8!
- b)4!
- c)(8!)(4!)
- d)7!(8P4)
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The number of ways of arranging 8 men and 4 women around a circular t...
The number of ways of arranging 8 men = 7!
The number of ways of arranging 4 women such that no two women can sit together = 8P4∴ Required number of ways =7!(8P4)
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The number of ways of arranging 8 men and 4 women around a circular t...
Arranging Men and Women around a Circular Table
To solve this problem, we need to find the number of ways to arrange 8 men and 4 women around a circular table such that no two women can sit together.
Step 1: Arranging the Men
Since the table is circular, we can fix one man at a position and arrange the remaining 7 men in a line. The number of ways to arrange 7 men in a line is given by 7!.
Step 2: Arranging the Women
To ensure that no two women sit together, we can place one woman between each pair of men. This creates 8 gaps where the women can be placed. Since there are 4 women, we can choose 4 gaps from the 8 gaps in 8P4 = 8!/(8-4)! = 8P4 ways.
Step 3: Combining the Arrangements
Once we have arranged the men and women separately, we can combine their arrangements. Since the table is circular, we need to account for the circular permutations. This can be done by dividing the total number of arrangements by the number of people, which is 12.
Therefore, the total number of arrangements is given by (7! × 8P4) / 12.
Step 4: Simplifying the Expression
Let's simplify the expression to get a clearer answer.
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
8P4 = 8 × 7 × 6 × 5 = 1680
12 = 2 × 2 × 3
Dividing 7! × 8P4 by 12, we get (5040 × 1680) / (2 × 2 × 3) = 7!(8P4).
Therefore, the correct answer is option D, 7!(8P4).
To solve this problem, we need to find the number of ways to arrange 8 men and 4 women around a circular table such that no two women can sit together.
Step 1: Arranging the Men
Since the table is circular, we can fix one man at a position and arrange the remaining 7 men in a line. The number of ways to arrange 7 men in a line is given by 7!.
Step 2: Arranging the Women
To ensure that no two women sit together, we can place one woman between each pair of men. This creates 8 gaps where the women can be placed. Since there are 4 women, we can choose 4 gaps from the 8 gaps in 8P4 = 8!/(8-4)! = 8P4 ways.
Step 3: Combining the Arrangements
Once we have arranged the men and women separately, we can combine their arrangements. Since the table is circular, we need to account for the circular permutations. This can be done by dividing the total number of arrangements by the number of people, which is 12.
Therefore, the total number of arrangements is given by (7! × 8P4) / 12.
Step 4: Simplifying the Expression
Let's simplify the expression to get a clearer answer.
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
8P4 = 8 × 7 × 6 × 5 = 1680
12 = 2 × 2 × 3
Dividing 7! × 8P4 by 12, we get (5040 × 1680) / (2 × 2 × 3) = 7!(8P4).
Therefore, the correct answer is option D, 7!(8P4).
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The number of ways of arranging 8 men and 4 women around a circular table such that no two women can sit together, isa)8!b)4!c)(8!)(4!)d)7!(8P4)Correct answer is option 'D'. Can you explain this answer?
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