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Seven women and seven men are to sit round a circular table such that there is a man on either side of every women; the number of seating arrangements is :
- a)(7!)2
- b)(6!)2
- c)6! x 7!
- d)7!
Correct answer is option 'C'. Can you explain this answer?
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Seven women and seven men are to sit round a circular table such that ...
6! x 7!
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Seven women and seven men are to sit round a circular table such that ...
To solve this problem, we can use the concept of circular permutations. Let's break down the problem into smaller steps:
Step 1: Arrange the men
Since there are 7 men, we can arrange them in a linear manner in 7! = 5040 ways.
Step 2: Arrange the women
Now, let's consider the women. Since there are also 7 women, we can arrange them in a linear manner in 7! = 5040 ways.
Step 3: Combine the arrangements
Now, we need to combine the arrangements of men and women. Since the table is circular, we need to consider the relative positions of men and women.
Let's place a man at the start of the table. We have 7 choices for the first man. Now, for each choice of the first man, we need to place a woman on either side of him. We have 7 choices for the first woman and 6 choices for the second woman.
Therefore, for each arrangement of men, there are 7 x 7 x 6 = 294 ways to arrange the women.
Step 4: Calculate the total number of arrangements
To get the total number of arrangements, we need to multiply the number of arrangements of men (5040) by the number of arrangements of women (294).
Total number of arrangements = 5040 x 294 = 1481760.
Step 5: Simplify the expression
To simplify the expression, we can rewrite 5040 as 7! and divide both the numerator and denominator by 7.
Total number of arrangements = (7! x 7 x 6) / 7 = 6! x 7!.
Therefore, the correct answer is option C) 6! x 7!.
Step 1: Arrange the men
Since there are 7 men, we can arrange them in a linear manner in 7! = 5040 ways.
Step 2: Arrange the women
Now, let's consider the women. Since there are also 7 women, we can arrange them in a linear manner in 7! = 5040 ways.
Step 3: Combine the arrangements
Now, we need to combine the arrangements of men and women. Since the table is circular, we need to consider the relative positions of men and women.
Let's place a man at the start of the table. We have 7 choices for the first man. Now, for each choice of the first man, we need to place a woman on either side of him. We have 7 choices for the first woman and 6 choices for the second woman.
Therefore, for each arrangement of men, there are 7 x 7 x 6 = 294 ways to arrange the women.
Step 4: Calculate the total number of arrangements
To get the total number of arrangements, we need to multiply the number of arrangements of men (5040) by the number of arrangements of women (294).
Total number of arrangements = 5040 x 294 = 1481760.
Step 5: Simplify the expression
To simplify the expression, we can rewrite 5040 as 7! and divide both the numerator and denominator by 7.
Total number of arrangements = (7! x 7 x 6) / 7 = 6! x 7!.
Therefore, the correct answer is option C) 6! x 7!.
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Seven women and seven men are to sit round a circular table such that there is a man on either side of every women; the number of seating arrangements is :a)(7!)2b)(6!)2c)6! x 7!d)7!Correct answer is option 'C'. Can you explain this answer?
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