Problem 10 - Calculus for Biology and Medicine

Question

Five people line up for a photograph. How many different lineups are possible?

Step-by-Step Solution

Answer

There are 120 different lineups possible.

Step 1: Understanding Factorials

In order to solve this problem, we need to understand what a factorial is. A factorial, denoted by the symbol '!,' represents the product of all positive integers up to a particular number. For example, 5! (read as 'five factorial') is equal to 5 × 4 × 3 × 2 × 1.

Step 2: Applying the Factorial to the Problem

To find the number of different lineups for five people, we can use the concept of permutations, which involves arranging a set of distinct items. The formula for finding the number of permutations of a set of 'n' items is n!, where 'n' is the total number of items. In this problem, we have 5 people, so we need to compute the factorial of 5, which is 5!.

Step 3: Calculating the Factorial

Now, let's calculate 5! step by step. Start by writing down the numbers from 5 to 1: 5, 4, 3, 2, and 1. Multiply them together as follows: \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]

Step 4: Conclusion

Hence, there are 120 different ways to arrange the lineup of the five people for the photograph. This solution applies factorial to count permutations for the lineup.

Key Concepts

Permutations

Permutations refer to the different ways in which a set of items can be arranged in order. It is a key concept in combinatorics and involves arranging all or part of a set of objects. When we talk about permutations, order matters. For example, arranging five people in a line is different from choosing any five people without caring about their order. To calculate permutations, we use a factorial, which multiplies a series of descending natural numbers. Example of Permutations:

• The sequence A, B, C is a different permutation from C, A, B.
• With five people, labelled as P1, P2, P3, P4, and P5, the permutation is calculated as 5! or 5 × 4 × 3 × 2 × 1.
• In this specific problem, you are finding out how many possible lineup arrangements exist for the people. The answer is 120 permutations.

Understanding permutations helps you recognize how to arrange objects systematically and is widely applicable in real-world situations, like organizing events or creating schedules. So, the next time you need to find the number of ways to arrange several objects, remember that order and factorials are key.

Combinatorics

Combinatorics is a branch of mathematics that focuses on counting, arranging, and finding patterns. It involves studying the way items can be selected or arranged and incorporates concepts like permutations and combinations. Elements of Combinatorics:

• Permutations: Concerned with the arrangement of items where the order does matter.
• Combinations: Focuses on the selection of items where the order does not matter.

In problem-solving, combinatorics provides a structured way to predict and calculate possibilities. For the exercise at hand, using factorials belongs to the permutation side of combinatorics because each person has a specific position, and changing just one person's spot changes the entire arrangement. Combinatorics is useful in areas ranging from computer science to probability theory. It forms the mathematical underpinning that makes counting and arranging every kind of data and possibility more straightforward. Whenever you approach a problem that involves arranging or selecting objects, you’re stepping into the world of combinatorics.

Mathematical Problem Solving

At its core, mathematical problem-solving involves identifying the right methods and tools to arrive at a solution. When solving a problem involving permutations and factorials, you’ll want to understand the whole process, from recognizing what’s being asked to arriving at the answer. Key Skills in Mathematical Problem Solving:

• Analysis: Break down the problem. Look at the information given and understand what’s required.
• Application: Apply relevant mathematical concepts, like permutations and factorials, fitting to the problem.
• Calculation: Perform any necessary computations with accuracy.
• Verification: Check if your solution makes sense and if you’ve correctly employed the concepts.

In the given problem, these skills involve recognizing that the arrangement of people needs permutations. You apply the factorial function to count these arrangements. Problem-solving in mathematics isn’t just about getting the right answer. It’s about setting a clear path to understanding that helps you hone skills that are useful in various aspects of life, from logical reasoning to analytical thinking.