Seating ArrangementsDate: 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/ |
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