In Exercises 5–36, express all probabilities as fractions.

Moving Company The United Van Lines moving company has a truck filled for deliveries to five different sites. If the order of the deliveries is randomly selected, what is the probability that it is the shortest route?

Deliveries are to be made at five different delivery sites.

The probability of selecting the shortest route needs to be calculated.

When a certain number of units are to be arranged in a queue, the formula of factorial is applied to computethe total number of possible arrangements.

Let A be the event of selecting the shortest route for delivery.

The total number of possible routes in which the deliveries can be made at the five sites is computed below:

$$\begin{gathered} 5!=5\times 4\times 3\times 2\times 1 \\ =120 \end{gathered}$$

The number of favorable ways in which the shortest route is selected is one.

The probability of selecting the shortest route is shown below:

$$\begin{gathered} P\left(A\right)=\frac{\text{Number}\text{of}\text{favourable}\text{ways}}{\text{Total}\text{number}\text{of}\text{ways}} \\ =\frac{1}{120} \end{gathered}$$

Therefore, the probability of selecting the shortest route is equal to $\frac{1}{120}$.