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Combinations : Seating Arrangements at a Dinner Table

In how many ways can 6 couples be seated at a circular table if each couple is not to be separated?

How many ways can 5 Manchester United and 8 Chlesea players be seated at a circular dinner table if no two Manchester United players can sit together?

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First of all conider that n people can be seated at a circular table in (n-1)! ways. That is because the first one can always choose one of the chairs, and then n-1 other people depending on that choice can choose the other chairs (consdiering the symmetry of the circular table).
After considering this, now we can solve the first part:

We have 6 couples, and these 6 couples want to be together when being seated. In such cases, we always consider the couples to be a set which has the same role of just one ...

Solution Summary

Combinations are used to find the number of ways of arranging people around a dinner table, subjest to specified conditions.