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Four children and four adults are to be seated at a circular table.  In how many different ways can they be seated if all the children are next to each other, and all the adults are next to each other?  (Two seatings are considered the same if one can be rotated to form the other.)

 Dec 23, 2023
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There are 4 children, which means there are 4! = 4 x 3 x 2 x 1 = 24 ways to seat the children.

There are also 4 adults, which means there are, again, 4! = 4 x 3 x 2 x 1 = 24 ways to seat the adults.

 

Since the children needs to sit next to each other, and all the adults needs to be next to each other, we can view all the children as 1 "object"and all the adults as 1 "object".

 

Now we need to find out how many ways we can place the childrens next to the adults.

children = C

adults = A

 

It can be:       CA                or                 AC

 

Since the problem said that two seatings are considered the same if one can be rotated to form the other, that means there is actually only ONE way to place them next to each other, since rotating AC gives us CA.

 

Finally, since we can seat the children in 24 different ways, and we can seat the adults in 24 different ways, that gives us 24 x 24 = 576 ways to seat the PEOPLES.

 Dec 23, 2023

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