Question

There are 4 different monitors and 6 different mother boards. How many different arrangements can be made to purchase a monitor and a motherboard? (a) 12 (b) 24 (c) 18 (d) \(6^{4}\)

Short answer

Answer: 24 arrangements

Step-by-step solution

Understand the problem

We are given 4 different monitors and 6 different motherboards. We must find the number of different arrangements to purchase a monitor and a motherboard.

Use the multiplication principle

Since we are given the number of different monitors and motherboards, we can find the total number of different arrangements by multiplying these two numbers. In this case, the multiplication principle states that the total number of different arrangements is equal to the product of the number of different monitors and the number of different motherboards.

Calculate the total number of arrangements

Now, we will calculate the total number of different arrangements by multiplying the number of different monitors (4) and the number of different motherboards (6). This can be calculated as follows: Total arrangements = 4 * 6

Find the answer

By calculating the total arrangements, we find: Total arrangements = 4 * 6 = 24 So, there are 24 different arrangements that can be made to purchase a monitor and a motherboard. The correct answer is (b) 24.

Key concepts

Multiplication Principle

The multiplication principle is a fundamental concept in combinatorics that helps determine the total number of possible outcomes when arranging multiple items or events.
It states that if one event can occur in "m" ways and another, independent event can occur in "n" ways, then the total number of ways both events can occur is the product of "m" and "n".

For instance, in the given problem, we have two distinct choices: selecting a monitor and selecting a motherboard.

• There are 4 different monitors
• There are 6 different motherboards

According to the multiplication principle, the total number of possible arrangements to make a purchase of one monitor and one motherboard is the product of these two numbers: \( 4 \times 6 = 24 \).
Thus, when faced with choices where decisions are made independently, the multiplication principle is a quick way to calculate the total number of arrangements.

Combinatorial Counting

Combinatorial counting is essential in determining how objects can be arranged or combined under different conditions. It involves using a set of techniques to count the possible configurations without having to list them explicitly.
This concept is vital in understanding problems related to arrangements, especially when dealing with permutations and combinations. In the case of choosing a monitor and a motherboard, we don't need to list all possible purchases to find the total number of arrangements; instead, we can rely on combinatorial counting principles.

• Calculate how many ways each element can be chosen (like our monitors and motherboards).
• Multiply these ways together if they're independent choices, as both choices do not affect each other.

Thus, we easily calculated the number 24 because arranging different monitors with different motherboards follows these guidelines. Combinatorial counting enables efficient problem-solving by reducing the complexity of counting tasks.

Elementary Probability Concepts

Elementary probability concepts provide the foundational basis for calculating the likelihood of different events occurring. These concepts often rely on methods like combinatorial counting to calculate probabilities effectively.
Probability, generally expressed as a fraction or percentage, represents how likely an event is to occur.

In our exercise with monitors and motherboards, if you wanted to choose one arrangement randomly and wanted to know the likelihood of picking any specific combination (e.g., a particular monitor and motherboard):

• Total combinations available: 24
• Probability of a specific arrangement: \( \frac{1}{24} \)

Since each arrangement is equally likely, the probability is uniform across all choices. Elementary probability builds on basic counting methods and applies them to understand the dynamics of randomness and choice. By mastering these concepts, one becomes capable of tackling more complex probability problems with confidence.