Question

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

Scheduling Routes A presidential candidate plans to begin her campaign by visiting the capitals of 5 of the 50 states. If the five capitals are randomly selected without replacement, what is the probability that the route is Sacramento, Albany, Juneau, Hartford, and Bismarck, in that order?

Short answer

The probability that the candidate’s route is in the given order is $\frac{1}{254251200}$.

Step-by-step solution

Given information

The presidential candidate is visiting fivecapitals of the 50 states of the United States as a part of her campaign.

The probability that the order will be from Sacramento to Albany to Juneau to Hartford to Bismarck is to be computed.

Define permutation

Permutation: The formula of permutation is used to compute the number of ways in which r units can be selected from n units without replacement in a particular order.

The formula is as follows:

${}^n{P}_r=\frac{n!}{\left(n-r\right)!}$

Calculation

The total number of ways in which five capitals can be selected from 50 states (when order matters) is computed below:

$$\begin{gathered} {}^{50}{P}_5=\frac{50!}{\left(50-5\right)!} \\ =\frac{50!}{45!} \\ =254251200 \end{gathered}$$

The number of favorable ways in which the given order of cities is followed is one.

The probability of visiting the cities in the given order is computed below:

$P\left(A\right)=\frac{1}{254251200}$

Therefore, the probability that the mentioned route in the order Sacramento, Albany, Juneau, Hartford, and Bismarck is $\frac{1}{254251200}$.