Discrete Math Topic - Combination/Permutation Let n be a positive integer. There

Question

Discrete Math Topic - Combination/Permutation

Let n be a positive integer. There are 4n passenger seats on a plane, and the seats arearranged in n rows of 4 seats each, where the 2nd and 3rd seats in each row are separated by the aisle. If 4n people are boarding the plane so that all passenger seats are lled, how manyseating arrangements are there, such that any 2 seating arrangements where every passengerhas the same neighbor is considered equivalent?

Note: Each person has exactly 1 neighbor, i.e. people in the 2nd and 3rd seats of each rowwho are separated by the aisle are not considered neighbors. People in the 1st seat of anyrow are neighbors with the people in the 2nd seat of the same row, and vice versa. Similarly,people in the 3rd seat of any row are neighbors with the people in the 4th seat of the samerow, and vice versa.

Explanation / Answer

Select pairs from 4n members and that can be done by

C(4n,2), ( number of ways os selecting 2 from 4n members )

And now each pair is considered as 1 unit and there are 4n/2=2n number of units which can be rearranged themselves. They can be arranged in (2n)! ways .

Hence the total number of ways can be written as

C(4n,2)*2n!

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