# In how many ways can four men and four women form a circle if the men and women alternate positions?

Four men and four women are forming a circle for a folk dance. In how many ways can this be done if we require that the men and women alternate positions?

My attempt was to square $24$ for total possibilities $= 576$. Then I tried to find out how many possibilities to subtract to find the answer. I got nowhere. I did find a couple of equations concerning circles and rings but they seemed contradictory:

1. The number of ways $n$ distinct objects can be arranged in a circle is $(n-1)!$.
2. The number of ways $n$ distinct beads (with $n\geq 3$) can be placed on a necklace is $(n-1)!/2$.

I don't see the difference.

Thank you for any insight you can give me.

T. Grode

• Well, I can flip a key ring over, so reversing the order isn't distinct ( so divide by two). I don't think you want to flip the dance floor over and consider the dancing arrangement unchanged. – Eric Towers Jun 5 '16 at 5:02
• @ArchisWelankar An answer without an explanation is not helpful. An incorrect answer without explanation, even moreso. It is downright harmful. – JMoravitz Jun 5 '16 at 5:09
• Let me check my mistake – Archis Welankar Jun 5 '16 at 5:11
• @ArchisWelankar It is a good idea to check your formulae against extreme cases to make sure that they work correctly. One should expect that with one man and one woman, there is only one way that they can stand in a circle. Make sure that your proposed formula does not give the answer of $2$. Further, with two men and two women, the only question is from one of the men's perspective which of the two women is on his right. There are only two arrangements in that case. – JMoravitz Jun 5 '16 at 5:16
• Ya now it's clear thanks sir. – Archis Welankar Jun 5 '16 at 5:38

## 1 Answer

Hint: Apply multiplication principle for the following steps:

• Arrange the men in a circle. Have the men leave a bit of extra space inbetween themselves.

• Once the previous step is completed, place a woman into each empty space.

The first step can be completed in $3!$ ways. The second step can be completed in $4!$ ways. Applying multiplication principle, there are then $3!\cdot 4!$ arrangements.

We can generalize this to $n$ men and $n$ women.

There will be $(n-1)!$ ways to do the first step. There will be $n!$ ways to do the second step. There will then be $(n-1)!\cdot n!$ total number of arrangements.

In counting how many options there are to complete the first step, we may recognize that we may divide by symmetry or we can come up with a more clever way of describing how to accomplish the first step without overcounting. For example:

• Set the man whose name appears first in alphabetical order on the dancefloor, it matters not where.
• Pick one of the remaining men to stand in clockwise position to the first man.
• Pick another of the remaining men to stand in the next available position clockwise.
• Repeat this process until all men are placed.
• that answer in incomplete, you must divide by 2 for eliminating redundancy of the other waywise order wich matches an arbitrary ordering, setting an individu m/f as an axis – Abr001am Jun 5 '16 at 13:35
• @Agawa001 no you don't. – JMoravitz Jun 5 '16 at 14:39
• i am sorry to say you must – Abr001am Jun 5 '16 at 15:43
• @agawa001 the two arrangements in your image are different. People standing in a circle are not necklaces. One arrangement obtained from another via rotations are considered the same. One obtained from another via a reflection are different. Imagine if we are doing a circular conga line traveling clockwise. If the person clockwise from me walks too fast it is of no discomfort to me. However if the arrangement were reflected, they would be stepping on my heels. The experiences are different, hence the arrangements are different. – JMoravitz Jun 5 '16 at 15:50
• @jm conga is done inline but here is a group circle dance as an irish or balkan dance – Abr001am Jun 5 '16 at 15:58