swerve wrote:At a circular table for eight will sit two children, their two parents, and the two parents of each of the children's parents, eight people in total. The children will sit together, the children's two parents will sit next to the children, one on either side of the two children, and the parents of each of the children's parents will sit next to each other beside the person who is their child. Given the restrictions, in how many ways can the people be arranged around the table?
A) 8
B) 16
C) 64
D) 96
E) 1440
To count circular permutations:
1. Place one person at the table.
2. Count the number of ways to arrange the REMAINING people.
After one of the two children has been placed at the table:
Number of options for the other child = 2. (To the left or right of the first child.)
Number of options for the mother of the two children = 2. (To the left or right of the two children.)
Number of options for the father of the two chidren = 1. (The remaining seat next to the two children.)
Number of options for the seat next to the mother = 2. (Either of her two parents.)
Number of options for the seat next to the mother's parent = 1. (Must be the spouse of the mother's parent.)
Number of options for the seat next to the father = 2. (Either of his two parents.)
Number of options for the remaining seat = 1. (Only 1 person left.)
To combine these options, we multiply:
2*2*1*2*1*2*1 = 16.
The correct answer is
B.
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