Question

True or False If a task consists of a sequence of three choices in which there are \(p\) selections for the first choice, \(q\) selections for the second choice, and \(r\) selections for the third choice, then the task of making these selections can be done in \(p \cdot q \cdot r\) different ways.

Short answer

True

Step-by-step solution

Understand the Problem

The task consists of making three choices. Each choice has a specific number of selections: the first choice has p selections, the second choice has q selections, and the third choice has r selections.

Identify Combinatorial Principle

This problem uses the fundamental principle of counting (multiplication principle). If there are p ways to do the first choice, q ways to do the second choice, and r ways to do the third choice, then the task can be completed by making these selections in p × q × r ways.

Apply the Principle

According to the multiplication principle, the number of different ways to perform the task is given by multiplying the selections: \( p \times q \times r \).

Conclusion

The statement given in the exercise aligns with the multiplication principle in combinatorics, hence it is true.

Key concepts

combinatorial principle

To understand the concept behind the given exercise, we first need to grasp the combinatorial principle. This principle is a foundational idea in combinatorics, which is a branch of mathematics focused on counting, arrangement, and combination of objects. The combinatorial principle helps us figure out the total number of ways to arrange a set of items by considering all possible choices.
For example, if you have three boxes and each box has multiple items to choose from, the combinatorial principle helps you determine how many different ways you can pick one item from each box. This idea is used widely in various fields such as computer science, statistics, and everyday life scenarios to solve problems involving choices and combinations.

multiplication principle

The multiplication principle is a specific type of combinatorial principle. It's also known as the fundamental principle of counting. This principle states that if you have a sequence of tasks, where each task has a set number of ways to be done, you can find the total number of ways to complete the sequence by multiplying the number of ways each task can be done.
Let's apply this to the given exercise. We have a task involving three choices: the first choice has p selections, the second has q selections, and the third has r selections. According to the multiplication principle, the total number of ways to complete this sequence of choices is given by multiplying the number of selections for each choice, which is \( p \times q \times r \).
This principle is very useful when dealing with multi-step processes where each step has different options, like planning a trip with different routes and stops, or creating a password with various character sets.

combinatorics

Combinatorics is the branch of mathematics dealing with combinations, permutations, and counting. It explores how different items can be arranged or combined following certain rules and constraints. Combinatorics has many real-life applications, such as in game design, scheduling, cryptography, and algorithm optimization.
In our exercise, we are dealing with a fundamental combinatorics problem: determining the number of ways to make three sequential choices. By using principles like the multiplication principle, we can easily solve these types of problems. For instance, if you need to arrange a playlist with selected songs from three different genres, combinatorics helps you calculate all possible arrangements.
By understanding the core concepts of combinatorics, you can tackle more complex problems involving larger sets and more restrictive conditions, making it a powerful tool in both academic and practical scenarios.