Question 334572: When seating guests at a round table, two arrangements are considered the same if each person has the same neighbor to the left and to the right in each arrangement. What are the number of unique arrangements when seating 2,3,4, and 6 guests at a round table.
Answer by Edwin McCravy(20055)   (Show Source):
You can put this solution on YOUR website! When seating guests at a round table, two arrangements are considered the same if each person has the same neighbor to the left and to the right in each arrangement. What are the number of unique arrangements when seating 2,3,4, and 6 guests at a round table.
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2 guests:
The seating arrangement
1. A, B
is the same as the seating arrangement
2. B, A
So when there are two people, we must divide the number of ways
they could stand in a line (2!) by 2. So the answer is    
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3 guests:
The seating arrangement
1. A, B, C
is the same as the seating arrangements
2. B, C, A,
and
3. C, A, B,
So when there are 3 people, we must divide the number of ways
they could stand in a line (3!) by 3. So the answer is    
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4 guests:
The seating arrangement
1. A, B, C, D
is the same as the seating arrangements
2. B, C, D, A
3. C, D, A, B
and
4. D, A, B, C
So when there are 4 people, we must divide the number of ways
they could stand in a line (4!) by 4. So the answer is    
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6 guests:
The seating arrangement
1. A, B, C, D, E, F
is the same as the seating arrangements
2. B, C, D, E, F, A
3. C, D, E, F, A, B
4. D, E, F, A, B, C
5. E, F, A, B, C, D
and
6. F, A, B, C, D, E
So when there are 6 people, we must divide the number of ways
they could stand in a line (6!) by 6. So the answer is    
Edwin
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