Problem 24 - Algebra and Trigonometry

Question

An automobile dealer offers five models. Each model comes in a choice of four colors, three types of stereo equipment, with or without air conditioning, and with or without a sunroof. In how many different ways can a customer order an auto from this dealer?

Step-by-Step Solution

Answer

There are 240 different ways to order an auto from this dealer.

Step 1: Identify choices for each category

Determine the number of options available for each category: models, colors, stereos, air conditioning, and sunroof. There are 5 models, 4 colors, 3 types of stereo equipment, 2 choices for air conditioning (with or without), and 2 choices for a sunroof (with or without).

Step 2: Calculate total combinations

Multiply the number of choices in each category to find the total number of configurations. The formula to use is: \(5 \times 4 \times 3 \times 2 \times 2\).

Step 3: Calculate the product

Perform the multiplication: \(5 \times 4 = 20\), then \(20 \times 3 = 60\), followed by \(60 \times 2 = 120\), and finally \(120 \times 2 = 240\).

Key Concepts

Permutations

Permutations refer to the various ways in which a set of items can be arranged. When we are concerned with permutations, the order of arrangement is crucial. This is different from combinations, where the order does not matter.
In the context of our exercise, although the primary focus is not directly on permutations, understanding this can help when considering scenarios such as arranging sequences of choices like models, colors, and stereo types in a specific order.

• When you choose a model, followed by a color, and so on, each selected sequence is a unique permutation.
• While the exercise itself ultimately uses a multiplication rule to find the total combinations, permutations come into play when considering each sequence as unique.

This understanding adds depth to considering how differently ordered selections can multiply the potential configurations.

Multiplication Rule

The multiplication rule is a fundamental principle in combinatorics used to calculate the total number of outcomes for a series of events.
When you have multiple options or steps, and you need to find the total number of combinations, the multiplication rule is your go-to strategy.
In our exercise involving the automobile dealer, each choice category is independent. Therefore, to find the total number of ways to configure a car, you multiply the number of choices available in each category.

• The 5 models: each one can be paired with any other choice.
• The 4 colors: independently chosen for each model.
• The 3 stereo types: each can accompany any model and color.
• The air conditioning choices: two possibilities for every previous combination.
• The sunroof options: also two choices for every combination.

The complete formula becomes: \(5 \times 4 \times 3 \times 2 \times 2 = 240\).
This principle simplifies calculating outcomes by breaking down complex configurations into stepwise multiplications.

Counting Principle

The counting principle is a straightforward and logical concept used to determine the number of possible combinations or outcomes. It simplifies scenarios where multiple independent options are involved.
The principle is analogous to breaking down a complex decision into multiple steps and assessing the number of choices at each step.

• In the given exercise: start by counting the number of models, colors, stereos, and so forth.
• Each decision, like choosing a model, doesn't affect the others, making the calculation of total configurations easier.
• Combining these choices involves multiplying their individual counts, as illustrated by the multiplication rule.

Thus, the counting principle helps you to "count" configurations and decide how different selections contribute to the overall number of arrangements.
It’s a key tool for solving problems like the automobile dealer's setup, ensuring no possibility is overlooked and helping provide a methodical approach to understanding combinations.