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 possible.

Step 1: Understanding the Problem

We have 5 cities we want to visit, and we need to find out in how many different ways we can arrange the order of visiting these cities. The task is a permutation problem because the order matters.

Step 2: Apply the Permutation Formula

The number of permutations of 5 distinct cities is calculated by finding the factorial of 5. In general, for any number of distinct items, n, the number of permutations is given by the formula \( n! \).

Step 3: Compute the Factorial

Calculate 5 factorial: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

Step 4: Simplify the Calculation

Perform the multiplication: \( 5 \times 4 = 20 \), then \( 20 \times 3 = 60 \), \( 60 \times 2 = 120 \), and finally \( 120 \times 1 = 120 \).

Step 5: Conclusion

Therefore, there are 120 different routes possible for visiting the cities of London, Paris, Amsterdam, Rome, and Heidelberg.

Key Concepts

Factorial Calculation

The factorial is a fundamental concept in permutation problems. It essentially helps us find out how many ways we can arrange a set of items. Factorial is denoted by an exclamation point. For a given number \( n \), \( n! \) (n factorial) is the product of all positive integers less than or equal to \( n \).
For example, to calculate \( 5! \), we multiply: 5, 4, 3, 2, and 1 together. The sequence for \( 5! \) is:

• 5 x 4 = 20
• 20 x 3 = 60
• 60 x 2 = 120
• 120 x 1 = 120

Factorial calculations are crucial for determining the number of permutations, especially when order matters, like in travel plans or seating arrangements.

Order of Arrangement

In permutation problems, the order of arrangement is key because changing the sequence of items creates a different outcome. For the European trip problem, the order in which you visit the cities is significant because it gives varying routes.
To comprehend why order matters, consider visiting two cities, Paris and Rome. The sequence of traveling to Paris first and then Rome is different from traveling to Rome first.
Thus, permutations focus on these sequences:

• London, Paris, Amsterdam, Rome, Heidelberg
• Paris, Amsterdam, Rome, Heidelberg, London
• Amsterdam, Berlin, Heidelberg, London, Paris
• ... and many more.

Each sequence or route counts as a unique permutation, hence knowing the total number of distinct sequences is crucial when planning.

Permutations Formula

The permutations formula is the primary tool used to calculate how many different ways a set of items can be arranged. When the order of arrangement is important, as in visiting cities, this formula is applied. The general permutations formula for \( n \) distinct items is given by \( n! \) (n factorial).
For five cities, our calculation would be:

• The formula is \( 5! \)
• This means we multiply: 5 x 4 x 3 x 2 x 1
• The result is 120 different orders.

Understanding and applying the permutations formula helps to easily solve problems where different arrangements or orders of a set are required, ensuring nothing is missed in planning or organizing.