In how many ways can 5 men and 5 women sit at a round table such that

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.

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!