## [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.
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 Recognitions: Homework Help Science Advisor 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?

 Quote by matt grime 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.

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