Associated Topics || Dr. Math Home || Search Dr. Math

### Seating Arrangements

```
Date: 02/28/99 at 23:16:31
From: Nicolas
Subject: Permutation and Combination Problem

Can you help me do the following problems?

a) A group of n married couples arrives at a dinner party and people
are seated around a circular table. The distance between a husband and
wife is defined to be equal to the number of people sitting between
them measured either clockwise or counter-clockwise, whichever gives
the smaller result. Considering all possible arrangements of the 2n
people, what is the average distance between the members of a
particular couple A?

b) Considering all possible seating arrangements for the 2n people,
what is the average number of couples, per arrangement, where both
members of the couple are seated side by side?"

Thank you.
```

```
Date: 03/01/99 at 16:27:31
From: Doctor Pat
Subject: Re: Permutation and Combination Problem

I thought at first that this would be messy, but it turns out pretty
neat. The average for all couples, by symmetry, is the same as the
average of any couple, so it is sufficient to study any one couple.
Let us rotate the table so that we have the husband seated. How many
places are there to seat his wife? Of course n-1, and of these two are
zero units away (next to him), two are one unit away, two are two units
away, and so on until she is directly opposite him, n-1 units away. So
the average distance is the product of

2*0+2*1+2*2+....+ 2*(n-2)+ 1* n-1

If you do this for a few couples you notice a simple relation:

couples                 average
2                       1/3
3                       4/5
4                       9/7
5                       16/9

n                       (n-1)^2/(2n-1)

Hope that helps.

- Doctor Pat, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search