Kenneth Rosen Edition7th Exercise 6.1 Example 20 (Page No. 394)

Question

what does it mean that "each person has same left and right neighbour "?

Comments

6 ??

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@ joshi_nitish pls explain answer yes 6 is the answer

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circular permuation is (n-1)!

Answer 1

Its a simple problem of circular permutation, how many ways to arrange 4 people in a circle i.e 3! = 6.
Now in question its mention two seating are considered same when each person has same right neighbor and same left neighbor, its simply means whatever arrangement we started with, if we get same arrangement that seating is considered same. Example

    A                  D
B    D  and  A       C  are consider same.
   C                   B

and in circular permutation anyhow these two arrangement are same. if you want to see what are  the 6 arrangement fix A at its position, arrange between BCD, you will get all 6 arrangement as BCD, BDC, CBD, CDB, DCB, DBC.

Answer 2

does it mean clock-wise and anti-clock-wise orders are not different

and so, Total number of circular-permutations  =  (n-1)!/2!

Ans = (4-1)!/2!= 3