Problem Solving

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

The OA will be provided later.

Solution:

Notice, the table is circular, but since every person to be seated has to be seated in relation to another person, and since the arrangement works from the center, where the two children are, to the outside, where the two sets of grandparents sit, the answer is the same as it would be were we arranging a row of eight people.

You can start this one in multiple ways. Maybe the best way is to start with the children.

The children can be arranged in 2 ways, C�C₂ or C₂C�.

The parents of the children have to sit next to the children and can also be arranged in 2 ways, P�CCP₂ or P₂CCP�.

The somewhat tricky part is understanding the implications of the fact that the pairs of grandparents are always at the sides of their respective children.

For instance, the grandparents that go with P� will always be next to P�, whether P� is to the left or to the right of the children.

We could make G� and G₂ the grandparents that go with P� and G₃ and G₄ the grandparents that go with P₂.

One possible arrangement is the following.

G�G₂P�C�C₂P₂G₃G₄

If P� and P₂ were to swap sides, their respective sets of grandparents would go with them.

G₃G₄P₂C�C₂P�G�G₂

So the positions of the pairs of grandparents are dictated by the positions of the parents that the grandparents sit beside.

Therefore the only way that the grandparents themselves can be arranged in different ways is by reversing them within their own pairs, for instance, the grandparents with P� could be arranged G�G₂P� or G₂G�P�.

So the parents and children can be arranged in 2 x 2 = 4 ways, and then for each of those four ways each pair of grandparents can be arranged in two different ways, giving us 2 x 2 = 4 ways to arrange the grandparents.

So the total number of ways in which they all can be arranged is 2 x 2 x 2 x 2 = 16 ways.

The correct answer is B.

Marty Murray 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

Strategy for circular permutations:
Place one person at the table.
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.