SOLUTION: 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 n

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.


---------------------------------------

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 

---------------------------------------

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 

---------------------------------------

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 

---------------------------------------

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

RELATED QUESTIONS
At a meeting, four scientists, two mathematicians, and a journalist are to be seated... (answered by ikleyn)
Twelve guests are to be seated for dinner, six of them have already chosen to sit in a... (answered by ikleyn)
A dinner party has 7 couples. The hostess wishes to seat all the guests around a circular (answered by edjones)
seven people are to be seated at the head table celebrating a 50th wedding anniversary.... (answered by VFBundy)
Ten people are to be seated at two circular tables. One table seats six people and the... (answered by ikleyn)
At a meeting, $4$ scientists, $3$ mathematicians, and $2$ journalists are to be seated... (answered by ikleyn,Edwin McCravy)
The same number of guests are seated at each of 11 large tables. There are 4 guests... (answered by Edwin McCravy)
if 4 Americans, 3 French men and 3 Ethiopians are to be seated in a row , how many... (answered by josmiceli)
Ten women and their husbands are to be seated on two sides of a long rectangular table,... (answered by jspindle)