Problem 23 - College Algebra with Modeling and Visualization

Question

A computer store offers a package in which buyers choose 1 of 2 monitors, 1 of 3 printers, and 1 of 4 types of software. How many different packages can be purchased?

Step-by-Step Solution

Answer

24 different packages can be purchased.

Step 1: Identify Choice Categories

The problem presents three categories of items to choose from: monitors, printers, and software. For each category, there is a specified number of choices.

Step 2: Determine Number of Choices per Category

According to the problem, there are 2 choices of monitors, 3 choices of printers, and 4 choices of software.

Step 3: Calculate Total Number of Packages Using Multiplication Principle

To find the total number of different packages, multiply the number of choices for each category: \(2 \times 3 \times 4\).

Step 4: Perform the Multiplication

Calculating the product gives \(2 \times 3 = 6\), and then multiply 6 by 4 to get 24.

Step 5: Verify the Calculation

Going through the multiplication again, \(2 \times 3 = 6\) and \(6 \times 4 = 24\), confirms that the total number of different packages is indeed 24.

Key Concepts

Combinatorics

Combinatorics, a branch of mathematics, is all about counting and arranging objects. It involves finding the number of ways things can be ordered or combined. When dealing with problems like choosing items from different categories, combinatorics provides the tools to figure out how many options are available. This can involve permutations, where order matters, and combinations, where it does not. In this case, we look at combinations because we are selecting items from different groups instead of arranging them in order. For example, when a computer store offers various monitors, printers, and software, you apply combinatorial techniques to determine all possible bundles a customer might select without needing to arrange these items.

Counting Principles

Counting principles are fundamental rules used to solve problems that involve determining the number of possible outcomes. One of the key principles is the Multiplication Principle, which is crucial in situations where you have several categories or stages of decision-making. The principle asserts that if one event can occur in "m" ways and a second event can occur independently of the first in "n" ways, then the two events together can occur in "m \times n" ways.

For the computer store package problem, we use this principle. We analyze each of the choices independently:

• 2 types of monitors
• 3 types of printers
• 4 software options

By multiplying the number of choices for each category: \(2 \times 3 \times 4\), we find the total number of distinct packages available. Understanding counting principles like these helps us systematically address complex problems by breaking them down into simpler parts.

Problem Solving in Algebra

Problem-solving in algebra often requires applying various rules and methods to find a solution. In counting problems, algebraic thinking consists of recognizing patterns and relationships among the different elements. Here, the problem is framed as finding the total number of unique product packages by analyzing each category's choices.

To tackle such problems effectively, it's essential to:

• Clearly understand the problem and identify relevant categories.
• Count the possible choices for each category.
• Apply multiplication for independent choices, using mathematical principles to arrive at the total count.

In this context, algebraic problem-solving involves expressing the relationships through mathematical expressions and ensuring calculations are accurate. This approach not only provides the solution but also enhances understanding of how algebra underpins many real-world problems.