Question

A businessman in New York is preparing an itinerary for a visit to six major cities. The distance traveled, and hence the cost of the trip, will depend on the order in which he plans his route. How many different itineraries (and trip costs) are possible?

Short answer

Answer: There are 720 different itineraries possible for the businessman to visit the six major cities.

Step-by-step solution

Find the number of cities visited

The businessman is preparing an itinerary to visit six major cities.

Determine the number of possible itineraries

In this problem, the order of the cities visited is essential. Using the formula for permutations, we can calculate the number of possible itineraries as n! (factorial) where n is the number of cities. Here, n = 6, so we have 6! potential itineraries for the businessman to visit the cities.

Calculate the number of permutations

Now, let's calculate the number of possible itineraries using the factorial formula: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 So, there are 720 different itineraries (and trip costs) possible for the businessman to visit the six major cities.

Key concepts

Factorial

Understanding the concept of a factorial is key to solving problems involving permutations. A factorial, represented as "\( n! \)," is the product of all positive integers up to a given number \( n \). It can be understood as the total number of ways to arrange \( n \) distinct items into a sequence. The standard formula for factorial is:

• \( n! = n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1 \)

For instance, \(6!\) means we multiply 6 by every whole number from 5 down to 1. This gives us \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).
The factorial function grows rapidly with increasing \( n \). Factorials help determine the number of possible sequences or arrangements of items and are fundamental in the study of permutations and combinations.Factorials are especially useful when the order of arrangement matters, which is often the case in real-world applications such as itinerary planning or determining possible outcomes in various scenarios.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arrangement, and combination of objects. When it comes to itinerary planning, combinatorics gives us the tools to calculate the number of possible routes and arrangements. By applying combinatorial techniques, we can systematically determine the number of ways to organize a sequence, especially when the order is crucial. In our example with the businessman visiting six cities, we used permutations to find the number of possible itineraries. This belongs to the area of combinatorics because we're interested in counting distinct sequences of cities. When using permutations, keep in mind:

• Order matters: The sequence in which the cities are arranged affects the outcome.
• Use factorials: Factorials give the number of different permutations for distinct items.

Understanding these combinatorial principles can aid not only in trip planning but also in solving other problems related to logistics, schedule organization, and beyond.

Itinerary Planning

Itinerary planning involves strategizing the order of visiting locations for efficiency, cost savings, or convenience. The choice in planning an itinerary can depend on factors like cost, distance, and time. In the given exercise, we focused on a businessman visiting six cities. It is crucial to consider that each different order of visiting these cities results in a different itinerary, influencing the trip's cost or duration due to varying distances. When planning an itinerary, consider:

• Priorities: Determine what is most important for your trip – time efficiency, cost, or visiting a specific sequence of places.
• Constraints: Are there specific limitations or requirements, such as fixed appointments?
• Optimization: Use mathematical techniques like permutations to understand the full range of possibilities.

By understanding permutations and applying combinatoric strategies, planners can develop multiple itinerary options, each with its own potential benefits, leading to the most optimal plan for their needs.