Question

how many ways can you arrange 10 people in a circle if two arrangements are considerded the same if each persons left and right neighbors are the same

Answer 1

1814400 ways

Explanation:

From the question, since it doesn't matter which seat you sit in as long as the neighbors either side of you still remain in the same order, thus;

Number of possible seat arrangements = 10! = 3628800

Now, we know that two seatings are the same when each person has the same two neighbors to the right or left. This simply means that it will be considered the same if the seats are placed around the circular table either clockwise or counter clockwise. With respect to this condition, we have to divide the number of possible seat arrangements by 2.

Thus;

Number of possible ways with the condition in the question = 3628800/2 = 1814400 ways