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Help I dont know this

 

Three adults and three children are to be seated at a circular table.  In how many different ways can they be seated if each child must be next to at least one adult?  (Two seatings are considered the same if one can be rotated to form the other.)

 Oct 3, 2023
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There are two cases to consider:

Case 1: One child sits next to two adults, and the other two children sit next to one adult each.

Case 2: Each child sits next to one adult.

In Case 1, we can choose which child sits next to two adults in 3 ways. Once we have chosen that child, we can seat the other two children and the three adults in a row in 6! ways. However, we have overcounted the number of arrangements by a factor of 2, since we have counted each arrangement as if the children were distinguishable.

In Case 2, we can seat the three adults in a row in 3! ways. Then, we can seat the three children between the adults in 3! ways. However, we have overcounted the number of arrangements by a factor of 3, since we have counted each arrangement as if the children were distinguishable.

Therefore, the total number of ways to seat the three adults and three children at a circular table if each child must be next to at least one adult is:

(3)(6!)/2 + (3!)(3!)/3 = 2! (6 + 3!) = \boxed{24}.

 Oct 4, 2023

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