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?
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?

Solution:
The total new employees =6
Therefore, they can be arranged in 6! Ways
\therefore \mathrm{n}(\mathrm{S})=6 !=6 \times 5 \times 4 \times 3 \times 2 \times 1=720
In 5 ways two adjacent desks for married couple can be selected that is (1,2),(2,3),(3,4),(4,5),(5,6)
In 2! Ways the married couple can be arranged in the two desks
In 4! Ways the other four persons can be arranged.
Therefore, the no. of ways in which married couple occupy adjacent desks

\begin{array}{l} =5 \times 2 ! \times 4 ! \\ =5 \times 2 \times 1 \times 4 \times 3 \times 2 \times 1 \\ =240 \end{array}
Therefore, no. of ways in which married couple occupy non-adjacent desks =6 !-240

\begin{array}{l} =(6 \times 5 \times 4 \times 3 \times 2 \times 1)-240 \\ =720-240 \\ =480=\mathrm{n}(\mathrm{E}) \end{array}

The required Probability =\frac{\text { No. of favourable outcome }}{\text { The total no. of outcomes }}
=\frac{\mathrm{n}(\mathrm{E})}{\mathrm{n}(\mathrm{S})}=\frac{480}{720}
=\frac{2}{3}