If n people are seated in a random manner in a row containin

If n people are seated in a random manner in a row containing 2n seats, what is the probability that no two people will occupy adjacent seats?

Solution

There are (2n)!/(n!n!) ways to choose seats for n people, and n! ways to permute them.

Hence, there are [(2n)!/(n!n!)]n! = (2n)!/n! ways to seat them.

There are only 2 ways to choose seats so that no two of them sit beside each other. There are n! ways to permute them after choosing the seats. Hence, there are 2(n!) ways to do the task.

Therefore,

P = [2(n!)] / [(2n)!/n!]

P = [2(n!)^2] / (2n)! [ANSWER]