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I need help here plz

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.)

Mar 19, 2023

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Because there are three children, this is the same as saying that the three children can’t sit next to each other since the child in the middle would have another child on both sides. So, to answer this, we can calculate

(total number of arrangements) - (arrangements with all three children together):

Part 1: Total Number of Arrangements

This is just six people seated around a circle which has (6 - 1)! = 5! = 120 possibilities.

Part 2: Total Number of Arrangements with All Three Children Together

Treat the children as a single group which means we’re arranging four groups around the table so:

(total arrangements of 4 items in a circle) * (number of ways to arrange the 3 children)

(4 - 1)! * 3!

3! * 3!

6 * 6

36

This makes the total number of possible arrangements 120 - 36 = 84.

Mar 19, 2023