Problem 56 - College Algebra with Modeling and Visualization

Question

There are 10 basic colors available for a new car, along with 5 basic styles of trim. In how many ways can a person pick the color and trim?

Step-by-Step Solution

Answer

There are 50 ways to pick the color and trim.

Step 1: Understanding Combinations

The problem requires us to find out how many different combinations of car colors and trims are possible. We have two choices to make: one for the color of the car and another for the trim of the car. For each color chosen, there are several trims available, hence it’s a combination problem.

Step 2: Identifying Number of Choices

Identify that there are 10 possible basic colors available for the car and 5 basic trim styles. This implies for every color chosen, there are 5 trim options available to pair with it.

Step 3: Calculate the Total Combinations

To find the total number of combinations of colors and trims, multiply the number of choices for colors by the number of choices for trims. The formula to use is: \( \text{Total Combinations} = \text{Number of Colors} \times \text{Number of Trims} \).

Step 4: Applying the Formula

Substitute the known values into the formula: \( \text{Total Combinations} = 10 \times 5 \).

Step 5: Calculate the Result

Perform the multiplication to find the total number of combinations: \( 10 \times 5 = 50 \). There are 50 possible combinations of color and trim choices.

Key Concepts

Basic Combinations

When we talk about basic combinations in the context of combinatorics, we refer to the counting of unique sets or groups. This is about determining the number of ways we can select or arrange items. In the car problem, each car color combined with a trim option forms a unique set or combination. We don't change the color or trim; we simply count how many combinations we can have.

• We have 10 basic colors to choose from for the car.
• Similarly, there are 5 different trim styles available.

For each color, you can match it with any of the 5 trims, creating a unique combination each time. This ability to pair each color with each trim exemplifies basic combinations in action.
By understanding the problem in terms of combinations, we can systematically count all the possible groupings, helping us solve the problem efficiently.

Multiplication Principle

The multiplication principle, often known as the rule of product, is a foundational concept in combinatorics. It's a method to determine the total number of possible outcomes for combined events. When we have multiple choices occurring, and each choice is independent of others, we multiply the number of options for each choice.
In our example, we have two independent choices:

• 10 car colors
• 5 trims

The multiplication principle tells us that to find the total number of color and trim combinations, we multiply these independent factors. Therefore, the formula used is:\[\text{Total Combinations} = 10 \times 5\]By following this rule, we calculate that there are 50 different ways to select a combination of car color and trim. Each color multiplied by each trim yields every possible variant.

Counting Problems

Counting problems in mathematics, and particularly in combinatorics, involve scenarios where we need to find the number of possible arrangements or selections. These problems can vary in complexity from simple cases to intricate puzzles involving several elements and restrictions.
For the car color and trim selection, we deal with a straightforward counting problem.

• We have clear and distinct groups: colors and trims.
• The task is to find how many ways we can select one color and one trim together.

Such problems are solved by employing basic principles like combinations and the multiplication principle.
Understanding counting problems requires breaking down the situation into understandable parts, determining what is being counted, and applying relevant mathematical principles to find the total number of outcomes. This systematic approach ensures that all possible options are considered, making it easier to manage and solve more complex scenarios in combinatorics.