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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   
Associated Topics:
High School Permutations and Combinations

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