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Bunuel
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In how many ways can 5 men and 5 women sit at a round table such that 03 Jan 2021, 15:41
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C
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In how many ways can 5 men and 5 women sit at a round table such that no two persons of the same gender sit next to each other?

A. 4!4!
B. 4!5!/2
C. 4!5!
D. 5!5!
E. 9!
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C
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Bunuel
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In how many ways can 5 men and 5 women sit at a round table such that 24 Jun 2023, 07:28
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jaykayes
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The number of arrangements of n distinct objects in a circle is given by (n-1)!.

Now, 5 men can be arranged in (5-1)! = 4!
We can place 5 women in empty slots between Men so that no women will be together and also no men will be together.
The number of arrangements of 5 women is 5!

The answer is 4!5!

Ans C


I understand how you did the first step, i.e., 5 men can be arranged in 4! ways around a circular table.

But in the next step, we have 5 empty slots between the 5 men.
These empty slots are also in a circular arrangement.
So, if 5 women are to be arranged in these 5 slots, the number of arrangements should also be 4!, right?
Therefore, the answer should be 4! x 4!, i.e., answer choice A.

I'm wondering how you got 5! for the second step... please explain!
Thanks in advance.
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  • The arrangement of n distinct objects in a row is denoted by n!.
  • The arrangement of n distinct objects in a circle is represented by (n-1)!.

The distinction between arranging in a row and in a circle is as follows: if we shift all objects by one position, we achieve a different arrangement in a row, but maintain the same relative arrangement in a circle. Therefore, for the number of circular arrangements of n objects, we have:

  • \(\frac{n!}{n} = (n-1)!\).

To demonstrate this, let's consider a simpler case: {1, 2, 3} arrangement in a row is distinct from {3, 1, 2} and from {2, 3, 1} arrangement.

However, the below three arrangements in a circle are identical:



Next, if we place three letters a, b, and c between these numbers, shifting them by one position would no longer yield the same arrangement. This is because it now matters whether a is positioned between 1 and 3 or between 1 and 2:



Therefore, the number of arrangements of a, b, and c between 1, 2, and 3 is now 3!, rather than 2!, due to the relative positioning of these letters in regards to the numbers.

Hope it's clear.

Show Spoiler
Attachment:
Arrangements in a cricle.png
Arrangements in a cricle.png [ 3.68 KiB | Viewed 6638 times ]

Attachment:
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Arrangements in a cricle2.png [ 9.21 KiB | Viewed 6599 times ]
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In how many ways can 5 men and 5 women sit at a round table such that 03 Jan 2021, 15:46
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The number of arrangements of n distinct objects in a circle is given by (n-1)!.

Now, 5 men can be arranged in (5-1)! = 4!
We can place 5 women in empty slots between Men so that no women will be together and also no men will be together.
The number of arrangements of 5 women is 5!

The answer is 4!5!

Ans C
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In how many ways can 5 men and 5 women sit at a round table such that 03 Jan 2021, 23:29
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5 men can arranged in circular arrangement in (5-1)! =4! ways
Now, 5 women have 5 places, so 5C5 ways and arranged in 5! ways.

So, Required combination = 4!* 5C5* 5! = 4! * 5!

So, I think C. :)
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In how many ways can 5 men and 5 women sit at a round table such that 24 Jun 2023, 06:12
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akadiyan
The number of arrangements of n distinct objects in a circle is given by (n-1)!.

Now, 5 men can be arranged in (5-1)! = 4!
We can place 5 women in empty slots between Men so that no women will be together and also no men will be together.
The number of arrangements of 5 women is 5!

The answer is 4!5!

Ans C
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I understand how you did the first step, i.e., 5 men can be arranged in 4! ways around a circular table.

But in the next step, we have 5 empty slots between the 5 men.
These empty slots are also in a circular arrangement.
So, if 5 women are to be arranged in these 5 slots, the number of arrangements should also be 4!, right?
Therefore, the answer should be 4! x 4!, i.e., answer choice A.

I'm wondering how you got 5! for the second step... please explain!
Thanks in advance.
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In how many ways can 5 men and 5 women sit at a round table such that 24 Jun 2023, 08:17
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After the women are placed using the circle principle (n-1)!, then there is a point of reference for placing the men, so they can be arranged in n! ways. You can start with either the women or the men, but one placement is (n-1)! and the next is n!
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In how many ways can 5 men and 5 women sit at a round table such that 25 Jun 2023, 03:16
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Bunuel
jaykayes
akadiyan
The number of arrangements of n distinct objects in a circle is given by (n-1)!.

Now, 5 men can be arranged in (5-1)! = 4!
We can place 5 women in empty slots between Men so that no women will be together and also no men will be together.
The number of arrangements of 5 women is 5!

The answer is 4!5!

Ans C


I understand how you did the first step, i.e., 5 men can be arranged in 4! ways around a circular table.

But in the next step, we have 5 empty slots between the 5 men.
These empty slots are also in a circular arrangement.
So, if 5 women are to be arranged in these 5 slots, the number of arrangements should also be 4!, right?
Therefore, the answer should be 4! x 4!, i.e., answer choice A.

I'm wondering how you got 5! for the second step... please explain!
Thanks in advance.

The arrangement of n distinct objects in a row is denoted by n!.
The arrangement of n distinct objects in a circle is represented by (n-1)!.

The distinction between arranging in a row and in a circle is as follows: if we shift all objects by one position, we achieve a different arrangement in a row, but maintain the same relative arrangement in a circle. Therefore, for the number of circular arrangements of n objects, we have:

\(\frac{n!}{n} = (n-1)!\).

To demonstrate this, let's consider a simpler case: {1, 2, 3} arrangement in a row is distinct from {3, 1, 2} and from {2, 3, 1} arrangement.

However, the below three arrangements in a circle are identical:



Next, if we place three letters a, b, and c between these numbers, shifting them by one position would no longer yield the same arrangement. This is because it now matters whether a is positioned between 1 and 3 or between 1 and 2:



Therefore, the number of arrangements of a, b, and c between 1, 2, and 3 is now 3!, rather than 2!, due to the relative positioning of these letters in regards to the numbers.

Hope it's clear.

Show Spoiler
Attachment:
Arrangements in a cricle.png

Attachment:
Arrangements in a cricle2.png
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Thank you, Bunuel, it's very clear now!
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In how many ways can 5 men and 5 women sit at a round table such that 25 Jun 2023, 03:18
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DanTheGMATMan
After the women are placed using the circle principle (n-1)!, then there is a point of reference for placing the men, so they can be arranged in n! ways. You can start with either the women or the men, but one placement is (n-1)! and the next is n!
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Thank you, DanTheGMATMan, that exactly answers my question!
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In how many ways can 5 men and 5 women sit at a round table such that 24 Oct 2025, 20:16
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