Question

Six new employees, two of whom are married to each other, are to be assigned six desks that are lined up in a row. if the assignment of employees to desks is made randomly, what is the probability that the married couple will have nonadjacent desks? (hint: first find the probability that the couple will have adjacent desks, and then subtract this number from 1.)

Answer 1

Here I'm using the hint, so I'm computing the probability that the two employees have adjacent desks. First of all we compute how many configurations are possible. Since there are 6 employees, the possible permutations (assignments to desks) are 6 factorial, i.e. 6 x 5 x 4 x 3 x 2 x 1 = 720. The couple could be in positions (1,2), (2,3), (3,4), (4,5), (5,6), and in every of these cases they can be in either order (so 5 x 2 cases) and in each of these cases the other 4 employees can be in any order (so 5 x 2 x 4 ! = 5 x 2 x 4 x 3 x 2 x 1) = 240. So, in 240 cases out of 720 the married employees are sit aside. Hence the probability we are looking for is 1 - 240/720 = 480/720 = 2/3 = 0.6666... = 66.666 %.