Problem 6 - Algebra for College Students

Question

A car can be ordered in any one of 8 different colors, with 3 different engine sizes, 2 different transmissions, 3 different body styles, 4 different interior designs, and 5 different stereos. How many different cars are available?

Step-by-Step Solution

Answer

The total number of different cars available is 2880.

Step 1 - Identify the choices

Identify the number of choices available for each category: colors, engine sizes, transmissions, body styles, interior designs, and stereos. For this problem: 8 colors, 3 engine sizes, 2 transmissions, 3 body styles, 4 interior designs, and 5 stereos.

Step 2 - Use the counting principle

According to the counting principle, if there are multiple categories and each category has a certain number of choices, the total number of combinations can be found by multiplying the number of choices for each category together.

Step 3 - Apply the counting principle

Multiply the number of choices together: \[ 8 \text{ colors} \times 3 \text{ engine sizes} \times 2 \text{ transmissions} \times 3 \text{ body styles} \times 4 \text{ interior designs} \times 5 \text{ stereos} \]

Step 4 - Calculate the total number of combinations

Perform the multiplication: \[ 8 \times 3 \times 2 \times 3 \times 4 \times 5 = 2880 \]So, the total number of different cars available is 2880.

Key Concepts

Counting Principle

The counting principle is a fundamental concept in combinatorics that allows us to determine the number of possible outcomes in a given situation. It is particularly useful when you need to calculate the total number of combinations of different choices or events. In essence, it states that if we have multiple categories (like colors, engine sizes, etc.), we can find out the total number of possible combinations by multiplying the number of choices in each category together.
For example, in the given exercise about car configurations, each category (colors, engine sizes, transmissions, body styles, interior designs, and stereos) has a specific number of choices. By using the counting principle, we calculate the total number of possible car configurations by multiplying these choices together.

Multiplication Principle

The multiplication principle is closely related to the counting principle and is simply a way to apply the counting principle through multiplication.
It helps us understand that if an event can occur in 'm' ways and another independent event can occur in 'n' ways, then the total number of ways both events can happen together is 'm' times 'n'.
In the car example, let's break it down:

• There are 8 possible colors.
• Each color can have 3 different engine sizes.
• Each color-engine combination can have 2 types of transmissions.
• Each color-engine-transmission combination can have 3 body styles.
• Each color-engine-transmission-body style can have 4 interior designs.
• Each color-engine-transmission-body style-interior design can have 5 stereos.

When we multiply all these choices together (8 * 3 * 2 * 3 * 4 * 5), we get the total number of configurations, which is 2880.
This exemplifies how the multiplication principle helps in determining the number of different combinations effectively.

Combinations

In combinatorics, combinations refer to the different ways of selecting items from a larger set. Unlike permutations, combinations do not consider the order of selection.
However, in the given car configuration problem, we need to consider all options, and each choice is dependent on the previous choices. That's why we use the counting principle combined with multiplication instead.
Therefore, when thinking about combinations in this context, we should focus on identifying and multiplying the available choices for each category to determine the total number of possible combinations.
This approach is very useful for problems where combination choices are being made across multiple categories, like coloring an object or setting up different features in a product.