Problem 36 - A Graphical Approach to College Algebra

Question

Use the fundamental principle of counting or permutations to solve each problem. Auto Varieties \(\quad\) An auto manufacturer produces 7 models, each available in 6 different colors, 4 different upholstery fabrics, and 5 interior colors. How many varieties of the auto are available?

Step-by-Step Solution

Answer

There are 840 different varieties of the auto available.

Step 1: Identify the Choices for Each Feature

The problem states that there are different choices for each model feature. Specifically, there are 7 different models, 6 different exterior colors, 4 upholstery fabrics, and 5 interior colors to choose from. We need to multiply these choices together to determine the total number of varieties.

Step 2: Apply the Fundamental Principle of Counting

According to the fundamental principle of counting, if there are multiple stages of choices, the total number of outcomes is the product of the numbers of choices at each stage. We calculate this as follows: \(7 \times 6 \times 4 \times 5\).

Step 3: Calculate the Total Number of Varieties

Multiply the numbers together to find the total number of varieties: \[7 \times 6 \times 4 \times 5 = 840.\]

Step 4: Interpret the Result

The calculation indicates that there are 840 different combinations of models, exterior colors, fabrics, and interior colors that the manufacturer produces. This represents the total number of different auto varieties available.

Key Concepts

Permutations

Permutations are a way of arranging or organizing a set of items into an ordered sequence. This concept is especially helpful when you want to understand how the order of items affects the outcome. For instance, if you're picking a first, second, and third place winner from a group of people, the order in which they are picked matters.
When working with permutations, it's important to remember:

• The total number of items available affects the number of possible arrangements.
• The order of selection matters, as each arrangement is unique.

In the context of the auto manufacturer problem, while order within the same feature (like color or model type) doesn’t matter, permutations become relevant when considering the sequence of choices available (i.e., selecting model -> color -> fabric -> interior color). The sequence of choosing a model, then the associated options, encapsulates this permutation concept. The step-by-step choices define a structured path where permutations guide the plan of action before the application of the fundamental counting principle.

Counting problems

Counting problems focus on determining the number of ways items can be combined or arranged in specific conditions. The Fundamental Principle of Counting (also known as the Counting Rule) is pivotal for solving these problems. It states that if you have multiple stages to choose from, and each stage has a fixed number of actions, then you multiply these choices to find the total number of outcomes.
In our exercise, we're tasked with finding how many different varieties of cars can be produced using the given options. There are several stages of choices, each representing a feature of the car:

• 7 models
• 6 colors
• 4 upholstery fabrics
• 5 interior colors

Our job is to multiply these numbers together. This application of the Fundamental Principle of Counting allows for an efficient solution to the problem by breaking it down into manageable parts.

Combinatorics

Combinatorics is the branch of mathematics that studies the counting, arrangement, and combination of objects in a set. It helps us solve problems related to counting without having to list every possible outcome. Combinatorics is foundational for solving permutation and combination problems.
In the given exercise about car varieties, we deal with a fundamental combinatorial concept: multiplying the number of choices. This doesn't involve complicated formulas or listing all outcomes. Instead, we utilize basic combinatorial logic: calculate the product of possibilities at each decision point.
Understanding combinatorics means recognizing:

• Not all choices need to be made at once; they build off each other.
• Effective strategies avoid repetition and unnecessary computation.

Therefore, combinatorics bridges the gap between basic counting problems and more complex arrangements, reinforcing the logical steps needed to decipher how many automobile varieties are possible, as demonstrated by the step-by-step solution applying these principles.