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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:

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

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 $\frac{1}{120}$.