Problem 35 - A Graphical Approach to College Algebra

Question

Use the fundamental principle of counting or permutations to solve each problem. Home Plan Choices 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 choices available

The builder offers three different types of choices for constructing a home: 5 basic plans, 3 roof styles, and 2 exterior finishes.

Step 2: Apply the fundamental principle of counting

According to the fundamental principle of counting, if there are multiple choices and each choice is independent, you multiply the number of possibilities for each choice. Here, you multiply the number of choices for plans, roof styles, and exterior finishes.

Step 3: Calculate the total number of combinations

The total number of different homes is given by multiplying the number of basic plans, roof styles, and exterior finishes: \(5 \times 3 \times 2 = 30\).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and analyzing distinct objects. It's the foundation for solving problems related to counting the possible arrangements or combinations of a set. In our problem, combinatorics helps us determine the total number of different homes available by systematically counting the various options. By breaking down options into manageable components, like home plans, roof styles, and exterior finishes, students can apply methods of combinatorics such as the fundamental principle of counting to calculate possibilities. This enables the enumeration of complex options efficiently, avoiding the need for listing all possibilities by hand. This principle is especially useful when choices are numerous and varied, just like in the home-building scenario.

Permutations

Permutations are specific arrangements of a set of objects. Unlike combinations that don't consider order, permutations do. However, in our home-building exercise, permutations are not directly applicable as the order of selecting home plans, roof styles, and exterior finishes doesn’t change the end product. But understanding permutations gives us insight into how various arrangements of a particular set, including order, are calculated. For example, if we had to determine the order in which a painter applied colors, permutations would be vital. It’s important to recognize when permutations are applicable – such as in arranging elements where sequence matters – versus scenarios like this where the selection of different components isn't sequential.

Independent Choices

Independent choices mean the selection in one category doesn’t affect the selection in another. This is crucial in applying the fundamental principle of counting. Here, selecting a basic home plan does not influence the choices available for roof styles or exterior finishes. This independence allows us to multiply the number of options in each category to find the aggregate number of different outcomes. For instance, having 5 plans, 3 roof styles, and 2 finishes, we simply multiply these numbers to calculate total possible combinations. Recognizing independence between choices helps solve complex problems by simplifying calculations, avoiding considerations of dependency, and making robust predictions about outcomes.