KENETH ROSEN

Question

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 .????

Comments

IN KENNETH ROSE ANSWER IS GIVENN AS 6

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can you post the screenshot of the given solution?

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i think it is(n-1)! ...6...because there are four people only then exchanging the place of any two person.....other left and right side person also get changed.....

Answer 1

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.

Comments

For the confusion, 

(a) If clockwise and anti clock-wise orders are different, then total number of circular-permutations is given by $\textbf{(n-1)!}$

(b) If clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by $(n-1)!/2!$

Here, the question explicitly mentions that left and right neighbours when interchanged have to be counted separately. This means that clockwise and anti clock-wise orders have to be counted separately.

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ok thannks

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i didn't get your comment, what do you mean by "clockwise and anti clock-wise orders are different/same"?
Please give a short example of five people sitting in such order. 

Answer 2

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. 

Answer 3

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