Counting Seating Arrangments of Couples at a Round Table

[e(ho0n3]

[SOLVED] Counting Seating Arrangments of Couples at a Round Table

I'm reading this example in my probability book which is I'm not understanding. It says:

There are 19! ways of arranging 20 people around a table. The number of arrangements that result in a specified set of n men sitting next to their wives can most easily be obtained by first thinking of each of the n married couples as being single entities. If this were the case, then we would need to arrange 20 - n entities around a round table, and there are clearly (20 - n - 1)! such arrangements.

There are 10 married couples by the way. The "20 - n entities" part is bugging me. Shouldn't that be 10 - n, given that there are 10 entities/married couples. I also don't understand how the (20 - n - 1)! part follows.

 

[matt grime]

Why ist it 10-n? If there is one married couple, there are 18 singletons, hence 19 objects to arrange (amazing what thinking of an example can do..). Thus if there are n couples hence how many single people? Now how many 'objects' are you arranging in a circle?

 

[e(ho0n3]

Thus if there are n couples hence how many single people? Now how many 'objects' are you arranging in a circle?

Technically, there are no single people (they're all in couples). However, if n of the couples have already been seated, then there are 20 - 2n seats around the table for the remaining 10 - n couples or 20 - 2n people.

 

[matt grime]

What does that show? (Apart from the fact that you seem to be focussed on the wrong thing.)

 

[e(ho0n3]

To be honest, I just don't understand the explanation. Here's how I would count the seating arrangements:

First, I would pick an ordering of the pairs and number them 1 through 20. Then I would sit pair 1 by first picking a seat for the woman and then picking a side for the man. This can be done in one of 20 * 2 ways.

Then I proceed with pair 2. The woman can sit in one of 18 ways. In 2 of those ways, the man is forced to sit in one spot. For the other 16 locations, the man can sit to the right or left. Hence, there are 2 + 16 * 2 ways to sit pair 2.

For pair 3, things get someone more complicated because I have to take into account of where pairs 1 and 2 are sitting. Ditto for pairs 4 - 10.

 

[matt grime]

Forget the married status or otherwise of the objects.

1. There are 20 objects,

2. we pair up 2n of them in n pairs.

3. We wish to arrange these n pairs and 20-2n remaining unpaired items in a circle.

4.That is we have 20-2n+n=20-n things to put in a circle

5. which can be done (20-n-1)! ways.

which of 1-5 is confusing?

Notice that the question does not distinguish between the order of the two objects in a pair, just that they are paired.

 

[e(ho0n3]

OK. I understand now. 3 had me confused because I thought it didn't make any sense to arrange paired and unpaired objects. Thank you.