Problem 50
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
Verified Answer
120 different skateboards can be constructed.
1Step 1: Understanding the Components of a Skateboard
A skateboard is typically made of three main parts: deck, trucks, and wheels. In this problem, we are given the number of types for each part. The shop has 10 types of board decks, 3 types of trucks, and 4 types of wheels available.
2Step 2: Analyze How to Combine Components
To form a complete skateboard, you must select one type of each component: one deck, one set of trucks, and one set of wheels. Each combination of these components will make a unique skateboard.
3Step 3: Determine the Total Number of Combinations
The total number of different skateboards that can be constructed is found by multiplying the number of options for each component. Using the Basic Principle of Counting, the total number of different skateboards is the product of the number of choices for each part.
4Step 4: Apply the Basic Principle of Counting
Calculate the total number of skateboards by multiplying the number of types for each part: \[10 \text{ decks} \times 3 \text{ trucks} \times 4 \text{ wheels} = 120 \text{ different skateboards}.\]
5Step 5: Conclusion
There are 120 different combinations of decks, trucks, and wheels. Thus, the shop can construct 120 different types of skateboards with the available components.
Key Concepts
Basic Principle of CountingMultiplication PrincipleProblem-Solving StepsTotal Combinations
Basic Principle of Counting
The Basic Principle of Counting is a fundamental method used in combinatorics to determine the total number of possible outcomes for a series of choices. It states that if you have a number of options for one event, and a number of options for a subsequent event, the total number of options for both events occurring together is the product of the number of options for each event. This principle is essential when you need to count the number of possible combinations of different items or events.
In the context of the skateboard shop example, the Basic Principle of Counting helps us understand how to determine the total number of unique skateboards that could be created from the available components. Since each part of the skateboard (deck, trucks, and wheels) has a certain number of choices, the principle guides us to multiply these numbers to find the total possible combinations of all parts.
In the context of the skateboard shop example, the Basic Principle of Counting helps us understand how to determine the total number of unique skateboards that could be created from the available components. Since each part of the skateboard (deck, trucks, and wheels) has a certain number of choices, the principle guides us to multiply these numbers to find the total possible combinations of all parts.
Multiplication Principle
The Multiplication Principle is closely related to the Basic Principle of Counting and is a specific rule in combinatorics that allows us to calculate the number of possible combinations of several independent choices. According to this principle, if an event can occur in 'm' ways and another independent event can occur in 'n' ways, then both events together can occur in \( m \times n \) ways.
When constructing a skateboard, for each type of deck choice, there are multiple choices of truck types and for each truck type, there are multiple choices of wheel types. This principle dictates that we multiply the number of choices for decks (10) by the number of choices for trucks (3), and finally by the number of choices for wheels (4) to obtain the total number of skateboard combinations. Thus, the calculation \(10 \times 3 \times 4 = 120\) uses this multiplication principle.
When constructing a skateboard, for each type of deck choice, there are multiple choices of truck types and for each truck type, there are multiple choices of wheel types. This principle dictates that we multiply the number of choices for decks (10) by the number of choices for trucks (3), and finally by the number of choices for wheels (4) to obtain the total number of skateboard combinations. Thus, the calculation \(10 \times 3 \times 4 = 120\) uses this multiplication principle.
Problem-Solving Steps
Effective problem-solving using the Basic Principle of Counting involves a series of logical steps. These steps help in organizing information and applying mathematical principles systematically. Here’s a breakdown of the applied problem-solving process in our skateboard shop example:
1. **Identify Key Components**: Recognize the parts needed for the combinations. Here, these are the skateboard deck, trucks, and wheels.
2. **Count the Options**: Determine how many choices are available for each component. The shop offers 10 decks, 3 trucks, and 4 wheels.
3. **Apply the Counting Principle**: Use the multiplication principle to calculate total combinations, simplifying complex choice scenarios into one understandable formula.
4. **Execute the Calculation**: Perform the multiplication: \(10 \times 3 \times 4\) to find that 120 unique skateboards can be assembled.
These steps clearly outline the logic from identifying options to executing calculations, ensuring clarity and comprehension.
1. **Identify Key Components**: Recognize the parts needed for the combinations. Here, these are the skateboard deck, trucks, and wheels.
2. **Count the Options**: Determine how many choices are available for each component. The shop offers 10 decks, 3 trucks, and 4 wheels.
3. **Apply the Counting Principle**: Use the multiplication principle to calculate total combinations, simplifying complex choice scenarios into one understandable formula.
4. **Execute the Calculation**: Perform the multiplication: \(10 \times 3 \times 4\) to find that 120 unique skateboards can be assembled.
These steps clearly outline the logic from identifying options to executing calculations, ensuring clarity and comprehension.
Total Combinations
Determining the total number of combinations is the ultimate goal when using the principles discussed. In many real-world scenarios, such as assembling products or planning events, understanding the number of possible combinations can significantly aid in decision-making.
The exercise with the skateboard shop beautifully illustrates this. Each selection of a deck, a set of trucks, and a set of wheels results in a distinct skateboard. By calculating the total combinations (as found earlier: \(10 \times 3 \times 4 = 120\)), you gain insight into the variety and inventory capability of the shop.
This computation is pivotal, not only in small-scale settings like this one but also in larger-scale scenarios, providing a foundational perspective on the flexibility and potential choices available in various situations.
The exercise with the skateboard shop beautifully illustrates this. Each selection of a deck, a set of trucks, and a set of wheels results in a distinct skateboard. By calculating the total combinations (as found earlier: \(10 \times 3 \times 4 = 120\)), you gain insight into the variety and inventory capability of the shop.
This computation is pivotal, not only in small-scale settings like this one but also in larger-scale scenarios, providing a foundational perspective on the flexibility and potential choices available in various situations.
Other exercises in this chapter
Problem 49
For the following exercises, graph the first five terms of the indicated sequence \(a_{1}=2, \quad a_{n}=\left(-a_{n-1}+1\right)^{2}\)
View solution Problem 50
For the following exercises, use this scenario: a bag of M\&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M\&Ms. Reaching into the bag,
View solution Problem 50
For the following exercises, use the information provided to graph the first five terms of the geometric sequence. \(a_{n}=27 \cdot 0.3^{n-1}\)
View solution Problem 50
For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\left\\{\frac{1}{3},-\frac{4}{3},-3, \ldots\right\\} $$
View solution