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How many different ways are there to seat four people around a circular table, where two seatings are considered the same when each person has the same left neighbor and the same right neighbor?

ANSWER IS 6 OR 3 .????

3 Answers

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Here when the right and left neighbours are interchanged, they have to be counted separately. Rotations of a given configuration are considered the same. These are criteria here based on which we have to calculate the permutations.

The easiest way would be to fix a given person and then we have $3! = 6$ ways of arranging the remaining positions. By doing this way, the repeated counting of rotations have been eliminated and left and right neighbours interchanged are counted separately.
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I think question has given condition to get the formula of circular permutation, watch the words "two seatings are considered the same when each person has the same left neighbor and the same right neighbor".

ABC    ACB
BCA    BAC
CAB    CBA
are 3!=6 different permutation of linear arrangement of ABC only 2 (i.e. 3!/3 or (n-1)! ) in cicular arrangement as (see vertically ) each person has the same left neighbor and the same right neighbor. 

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All the guests can sit around the table in 4!=24 ways. Since two seatings are considered equal if they all have the same neighbor, the location of the first person, does not matter. Hence, a division by 4. Also, since the left/right orientation is not an issue, we can 'flip' (place the first person at a seat, and instead of adding persons to the right, do it to the left) the seating and thus divide by 2. This gives 3 seatings in total.

Source: https://math.stackexchange.com/questions/3877764/how-many-ways-are-there-to-seat-six-people-around-a-circular-table/3878515

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