Problem 9 - Calculus for Biology and Medicine

Question

You plan a trip to Europe during which you wish to visit London, Paris, Amsterdam, Rome, and Heidelberg. Because you want to buy a railway ticket before you leave, you must decide on the order in which you will visit these five cities. How many different routes are there?

Step-by-Step Solution

Answer

There are 120 different routes to visit the five cities.

Step 1: Identify the Problem

You need to find out how many different ways you can visit 5 cities: London, Paris, Amsterdam, Rome, and Heidelberg, each exactly once. This is a permutation problem, as the order of visiting cities matters.

Step 2: Recognize the Permutation Formula

The number of ways to arrange 5 items is given by the factorial of 5, written as \( 5! \). Factorials are used in permutations to calculate the total possible arrangements.

Step 3: Calculate Factorial of 5

To calculate \( 5! \), multiply the numbers from 5 down to 1: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

Step 4: Compute the Product

Perform the multiplication: \( 5 \times 4 = 20 \); \( 20 \times 3 = 60 \); \( 60 \times 2 = 120 \); \( 120 \times 1 = 120 \). Thus, \( 5! = 120 \).

Key Concepts

Understanding Factorials

The concept of factorial is key to solving many permutation problems. A factorial, often denoted with an exclamation mark (!), refers to the product of an integer and all the integers below it. For example, the factorial of 5, written as \( 5! \), is calculated by multiplying 5 by every positive integer less than 5. Mathematically, you represent this as \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).
Factorials allow us to determine how many ways we can arrange a set number of items into a unique order. As the number increases, the factorial value grows very quickly. For instance, while \( 5! \) results in 120 different arrangements, \( 6! \) increases this to 720, showing how factorials capture the complexity of larger sets.
In permutations, factorials are used when every item needs a distinct position, as in the case of planning a unique travel route.

Exploring Combinatorics

Combinatorics is the area of mathematics that deals with counting, arrangement, and combination of objects. It helps us tackle problems like how many ways we can arrange or select items. Combinatorics is divided mainly into permutations and combinations.
Permutations focus on the arrangement of items where the order is important. For instance, if you are arranging your city visits, changing the order of cities results in a different itinerary, which is a permutation. The total number of permutations with all items being arranged is given by the factorial of the number of items.
Combinatorics is a powerful tool in mathematics because it provides methods to figure out problems that can be too challenging to list out manually, especially for larger numbers.

Arranging Items in Order

When talking about arranging items, especially in permutation problems, the order of placement is what matters most. The sequence in which items or steps are arranged can change the nature of the problem entirely. For a group of items, the number of possible arrangements increases factorially with the number of items.
For example, arranging five cities in a travel plan involves calculating how many unique sequences can be formed. Using the factorial function, you determine that there are 120 different sequences from those five cities, as each sequence is a unique permutation.
This concept is crucial in scenarios where order dictates the outcome or when each arrangement yields a different result, like crafting schedules, allocating resources, or planning itineraries.