Problem 35 - A Graphical Approach to Precalculus with Limits

Question

Use the fundamental principle of counting or permutations to solve each problem. How many different types of homes are available if a builder offers a choice of 5 basic plans, 3 roof styles, and 2 exterior finishes?

Step-by-Step Solution

Answer

There are 30 different types of homes available.

Step 1: Identify the Options

Identify the different choices given in the problem. The builder offers 5 basic plans, 3 roof styles, and 2 exterior finishes.

Step 2: Apply Fundamental Principle of Counting

Use the fundamental principle of counting, which states that if there are multiple choices, the total number of combinations is the product of the number of choices available at each step.

Step 3: Calculate Total Combinations

Multiply the number of choices for each aspect of the home: 5 basic plans, 3 roof styles, and 2 exterior finishes. The calculation is: \(5 \times 3 \times 2\).

Step 4: Compute the Result

Perform the multiplication to find the total number of different home types: \(5 \times 3 = 15\), and \(15 \times 2 = 30\).

Key Concepts

Permutations

Permutations are all about understanding arrangements where order matters. This concept is an essential part of combinatorics and is often encountered when figuring out all possible ways to arrange a set of items. For example, if you have three different books, and you want to arrange them on a shelf, each different order counts as a unique permutation.
You can think of permutations as rearranging objects where each specific sequence is important, unlike combinations where order does not matter.
To calculate permutations, you would use the formula:

• \( n! \) (n factorial) which is the product of all positive integers up to n

This means if you have a set of items you want to arrange, you multiply them together as follows: \( n \times (n-1) \times (n-2) \times ... \times 1 \).
Permutations become particularly useful when you need to solve problems involving ranking, schedules, or any cases where items are subject to specific sequences.

Combinatorics

Combinatorics is the broad mathematical topic that deals with counting, arrangement, and selection. It's like the art of arranging objects under specific rules and sets. You'll often use the concepts of permutations and combinations when you're diving into combinatorics.
In simpler terms, combinatorics answers questions like 'How many ways can a task be completed?' or 'How many different groups can be formed from a larger set?'. It's handy in various fields like computer science, statistics, and more.
There are key topics within combinatorics such as:

• **Permutations** – Considering all possible arrangements of a set
• **Combinations** – Focused on selecting subsets
• **The Fundamental Principle of Counting** – Which forms the basis for calculating possibilities when choices are independent

By mastering combinatorics, you can solve complex problems easily, especially when dealing with large sets or multiple conditions.

Multiplication Rule

The Multiplication Rule, often referred to as the Fundamental Principle of Counting, is pivotal in solving numerous problems. This rule helps to determine the total number of outcomes in a scenario where there are several stages or events, each with different choices.
When you have multiple independent choices to make, such as picking a type of bread, a type of filling, and a type of topping for your sandwich, the Multiplication Rule states that you simply multiply the number of choices available at each step.
Consider a simple scenario to understand how it works:

• You have 3 choices for breakfast.
• 4 choices for lunch.
• 2 choices for dinner.

To find out how many different meal plans you can create for a day, multiply the number of choices for each meal: \(3 \times 4 \times 2\).
This principle is straightforward but extremely powerful in fields ranging from computing all variations in genetic sequences to planning different air travel routes.