Problem 49 - College Algebra

Question

A skateboard shop stocks 10 types of board decks, 3 types of trucks, and 4 types of wheels. How many different skateboards can be constructed?

Step-by-Step Solution

Answer

There are 120 different skateboards possible.

Step 1: Understand the Components of a Skateboard

A skateboard is made up of three main components: the deck, trucks, and wheels. From the problem, the shop has 10 types of decks, 3 types of trucks, and 4 types of wheels.

Step 2: Calculate the Total Combinations for One Skateboard

To find the number of different skateboards that can be constructed, multiply the number of options available for each component. This is based on the rule of multiplication for combinations: \[\text{Total combinations} = (\text{Number of decks}) \times (\text{Number of trucks}) \times (\text{Number of wheels})\]

Step 3: Plug in the Values

Using the numbers provided: \[= 10 \times 3 \times 4\]

Step 4: Perform the Multiplication

First, multiply the number of decks by the number of trucks:\[10 \times 3 = 30\]Then, multiply the result by the number of wheels:\[30 \times 4 = 120\]

Step 5: Final Answer

Therefore, there are 120 different combinations of skateboards that can be made with the given components.

Key Concepts

Multiplication Principle

The multiplication principle is a fundamental concept in combinatorics. It is often used to calculate the total number of possible outcomes by multiplying the number of choices available at each step. Consider the example of constructing a skateboard, where you have multiple options for decks, trucks, and wheels.
Here are some key points to remember:

• Each choice is independent of the others.
• If you make a choice at each step, the total number of combinations is the product of all the individual choices.

This principle is not just limited to skateboards. It applies to any scenario where you have several independent choices, each with multiple options.

Combination Calculation

Combination calculation refers to the process of finding the total number of ways different items can be combined. Using the multiplication principle, this becomes straightforward when the choices are independent.
For a skateboard shop offering 10 decks, 3 trucks, and 4 wheels:

• Calculate the total number of combinations by multiplying the number of options available for each component.
• The formula used here is: \[\text{Total combinations} = (\text{Number of decks}) \times (\text{Number of trucks}) \times (\text{Number of wheels})\]
• By plugging in the numbers: \[10 \times 3 \times 4 = 120\]

This method efficiently gives the total number of different skateboards that can be constructed.

Problem Solving

Problem solving in combinatorics often involves breaking down the situation into manageable parts, like identifying the components involved. In this skateboard example, understand that each part of the skateboard—deck, trucks, and wheels—represents a separate choice.
Approach each problem step-by-step:

• Identify the components involved in making a choice.
• Apply the multiplication principle to calculate the total combinations.
• Check your calculations to ensure accuracy.

Problem solving is about methodically applying known principles, like the multiplication principle, to come to a solution. Practicing with different scenarios will help build confidence and improve your skills in solving combinatorial problems.