# Counting - seat men and women alternating at a table.

The problem:

Six men and six women are to be seated around a table, with men and women alternating. The chairs don't matter only who is next to whom, but right and left are different. How many seating arrangements are possible?

My initial approach is to think of it as pairs of boxes and find the way in which you can split the pairs and then multiply by 2 since left and right are not the same.

visually if $x$ is an empty spot to be taken with | denoting a boundary:

x x | x x | x x | x x | x x | x x


there are 11 gaps, if each boundary is put in between each x and you want to pick 2. Because chairs don't matter, I thought it might be:

$${11! \over {2!(11-2)!}} * 2$$

• Often I will point out what the mistaken number actually counts, but I'm at a loss for what your number describes in this scenario... my best would be that it counts the number of ways that you can line up twelve red marbles, a black marble, and a blue marble, such that the black marble and blue marble aren't adjacent and the furthest left and furthest right marbles are both red. (red marbles being your x's, the black and blue marble being your two gaps chosen between the x's, and the fact that black and blue marbles different colors accounts for your extra factor of 2). Seems far removed – JMoravitz Aug 5 '17 at 4:01

Seat Albert anywhere. Then for the other chairs, you know the gender of the person that will sit in it.

There are $5!$ ways to place the remaining men in the "man" seats.

Then there are $6!$ ways to seat the women in the "woman" seats.

So there are $6!5!$ total arrangements.

• Now I'm confused... your answer made me re-read the problem and these phrases "chairs don't matter" and "but right and left are different" seem to contradict one another. – FakeBrain Aug 5 '17 at 4:01
• @FakeBrain in this case, "chairs dont matter" meaning we don't care who sits specifically in the northernmost chair or in the westernmost chair, but rather we care specifically about who is to who's right and left. I.e. any arrangement around the table is considered the same as any other arrangement around the table where the difference is only rotating the seats. "Right and left are different" means that only rotation maintains "sameness", but reflection does not. I.e. we care who is to the right of albert differently than who is to the left. – JMoravitz Aug 5 '17 at 4:04
• @FakeBrain if "right and left weren't different" to us, we care only about who was adjacent to who, but not specifically which side they were on. E.g. if we cared only about who could have conversations with who at the table if people talk only to their neighbors, right and left wouldn't matter so much... but if they were eating and potentially bumping elbows as they used their forks and knives right and left would certainly cause a difference in experience. – JMoravitz Aug 5 '17 at 4:05