*sort by* **best** latest

### Best Answer Aficionada says

I have an answer, and I have an explanation, but it may be pretty far off, and I hate the fact that I can't come back later with another answer if I think about this some more and come up with something better!

First, my answer is 72 - if I understand the concept of a "free arrangement" correctly. As I understand it, a mirror image (or an identical sequence in reverse) would not count as a separate arrangement. Nor would a sequence count separately, if it is the same as another sequence, but simply starting at a different point.

Normally, in calculating simple permutations, we take the number of possibilities for the first choice, for the second choice, third choice, etc., and multiply them together. If that were the case here, we would multiply 4 choices for the first person (ladies first), 4 for the second (must be a man), 3 for the third (a woman), 3 for the fourth, 2 for the fifth (again, a woman), and 2 for the sixth, leaving only one man and one woman to fill the final places. With those possibilities, the permutations (non-circular) would equal 4*4*3*3*2*2=576.

However, since mirror images are not allowed (I think), and since any sequence or permutation can have one and only one exact mirror image or numerical palindrome, we have to divide the result by two, to eliminate those repetitions. 576/2 = 288.

What about eliminating the rotations? Well, if I understand correctly, a rotation is actually simply the same thing as starting an existing sequence at a different point and continuing it until you get back to the same point from which you started. In that case, it doesn't really matter which person is the first in order, and so instead of four possibilities for the first choice, we basically only need one. Once we can create one set of permutations using any one of the possibilities first, then all of the others (the non-free arrangements) could be found by starting with one of the other three options in the permutations that have been discovered. So, to eliminate those choices, divide by 4, effectively changing the first possibility to only 1. So, 288/4 = 72. That is how I came up with 72.

Then I checked my work. I did. I wrote out all 72 permutations to see whether I had overlooked some duplicates. So far, none. To make it easier on myself, I used letters A-D for one gender and digits 1-4 for the other.

I'll be interested to see whether I understood the problem correctly and/or whether I have overlooked something important.

It's a good sign when two people get the same answer using different methods. You beat me by two minutes, lol.

- See all 4 comments

### paxwill says

I get 72.

First deal with the women. Fix one woman in a chair. There are 3 choices for whom to seat directly across from her. Where you seat the other two women in the two remaining chairs is irrelevant since one arrangement will be a reflection of the other, and reflections are not counted as distinct. So there are 3 fundamentally different ways to arrange just the four women in every other chair.

Now deal with the men. Since the women are fixed, there are 4 ways to seat the first man, 3 ways to seat the second, 2 ways to seat the third, and 1 way to seat the last. Even if some of the arrangements of men are reflections or rotations of each other with respect to just the men, they will be distinct placements with respect to the already seated women. There are 4! = 24 ways to arrange the men for any given arrangement of females.

Since 3*24 = 72, that is the total number of distinct free seating arrangements.

* * * * *

Alternatively, you can count the total number of arrangements where reflections and rotations are considered distinct, which gives you 2*4!*4! = 1152. If you divide this by the size of the dihedral group for an octagon (16), you get 1152/16 = 72.

This last part is over my head for now. Maybe you could point me towards a Hub or other article that can help me to understand it?

- See all 3 comments