Question

In planning a round trip from Cleveland to Dover by way of New York, a traveler decides to do the Cleveland-New York segments by air and the two New York-Dover segments by steamship. If six airlines operate flights between Cleveland and New York and four steamship lines operate between New York and Dover, in how many ways can the traveler make the round trip without using the same company twice?

Short answer

360 ways.

Step-by-step solution

Understanding the problem

The traveler wants to make a round trip from Cleveland to Dover through New York. He will fly from Cleveland to New York (and back) using airlines and take a steamship from New York to Dover (and back). The traveler cannot use the same company twice during the trip.

Determine choices for the flight from Cleveland to New York

There are 6 airlines available for the flight from Cleveland to New York. The traveler can choose any one of these 6 airlines for the first leg of the trip.

Choose an airline for the return flight

Once the airline to New York is chosen, the traveler cannot use the same airline for the return trip. Therefore, after selecting an airline for the flight to New York, there are 5 remaining airlines available for the return flight to Cleveland.

Determine choices for the steamship from New York to Dover

There are 4 steamship lines available for travel from New York to Dover. The traveler can choose any one of these 4 steamship lines for this segment.

Choose a steamship for the return trip

Once a steamship line is chosen for the New York to Dover segment, the traveler cannot use the same steamship line for the return trip from Dover to New York. Thus, there are 3 remaining options for the return trip.

Calculate the total number of ways

To find the total number of possible ways to make the trip, multiply the number of available options for each segment: 6 choices (to New York) x 5 choices (return to Cleveland) x 4 choices (to Dover) x 3 choices (return to New York) = 360 possible ways.

Key concepts

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics. They help us determine different ways to arrange or select items, usually without repeating options. In the context of travel route planning, these concepts allow us to identify all possible ways a trip can be arranged. For example, if a traveler plans a round trip but does not want to use the same company for two segments, permutations are used to calculate the distinct sequences of travel choices.

To understand permutations, consider the choices for the flights in our scenario. The traveler initially has 6 airlines to choose from for the Cleveland to New York leg. Once one airline is chosen, only 5 remain for the return trip. This is because permutation handles order and considers each possible line-up of airlines as unique.

In contrast, combinations would only focus on selecting groups without particular order, which is not applicable here due to the sequential nature of the trips. Thus, permutations are the best fit to solve this situation where the order of choices matters.

Travel Route Planning

Travel route planning involves choosing the best possible paths and methods to reach a destination, considering constraints such as not using the same company twice in a trip. In our exercise, the traveler is planning a round trip from Cleveland to Dover through New York. This journey includes segments by air and steamship, which necessitate careful planning to ensure distinct choices are made.

The trip is divided into four main segments:

• Flying from Cleveland to New York
• Flying back from New York to Cleveland
• Sailing from New York to Dover
• Returning from Dover to New York

These segments must use different companies, as stated in the exercise. The challenge in travel route planning lies in selecting different providers for each leg, making sure the overall experience is efficient and meets all constraints set by the traveler.

Additionally, successful travel planning often requires considering other factors like schedules, costs, and personal preferences. But, for our mathematical approach, the focus is purely on calculating the number of ways the round trip can be completed given the conditions.

Discrete Mathematics

Discrete mathematics is a branch of mathematics dealing with discrete elements and has applications in computer science, cryptography, and logistics. It covers a range of topics, including set theory, graph theory, and, as in this case, combinatorics. This field is essential in solving problems that involve discrete, individual steps or items, unlike continuous functions.

In the travel example, discrete mathematics helps solve how many ways a traveler can complete a journey under certain constraints. The process of multiplying options for each leg of the trip (6 flight options to New York, 5 for the return, 4 steamships to Dover, 3 return) is a direct application of principles from this field.

The key to using discrete mathematics effectively is breaking a large problem into smaller, manageable parts. Each part can be analyzed using set rules, such as not repeating options, to derive a final answer. This approach is precise and follows logical steps to solve complex problems like our travel route example.