Problem 44 - Algebra 2

Question

Darius can do his homework in pencil or pen, using lined or unlined paper, and on one or both sides of each page. How many ways can he do his homework?

Step-by-Step Solution

Answer

Darius can do his homework in 8 different ways.

Step 1: Identify Choices for Writing Tool

Darius has the choice between using a pencil or a pen. Thus, there are 2 options for the writing tool.

Step 2: Identify Choices for Paper Type

He can choose between lined paper and unlined paper for his homework. This presents him with 2 options for paper type.

Step 3: Identify Choices for Page Sides

Darius can choose to write on one side of each page or on both sides. This gives him 2 options for how the pages are used.

Step 4: Calculate Total Number of Combinations

To find the total number of ways Darius can complete his homework, multiply the number of options for each decision: Writing tool (2) × Paper type (2) × Page sides (2). This results in 2 × 2 × 2 = 8 combinations.

Key Concepts

Multiplication principle

The multiplication principle is a fundamental concept in combinatorics. It helps us understand how to find the total number of possible outcomes when there are multiple independent choices to be made.

Imagine you are picking an outfit and you have 2 shirts, 3 pants, and 2 pairs of shoes to choose from. To find the total number of different outfits you can create, you'll apply the multiplication principle by multiplying the number of options for each choice:

• Shirts: 2 options
• Pants: 3 options
• Shoes: 2 options

The total number of outfits is calculated as: \( 2 \times 3 \times 2 = 12 \).

In the context of Darius's homework scenario, he makes three distinct decisions (writing tool, paper type, and page sides), each with 2 possibilities. Thus, the multiplication principle tells us that the total number of homework combinations is \( 2 \times 2 \times 2 = 8 \). Understanding this principle provides the tools needed to examine combinations in varied tasks and scenarios.

Decision-making process

The decision-making process in combinatorics involves understanding and listing all possible choices for each option. These options must be independent to apply the multiplication principle effectively.

In Darius's homework exercise, the decision-making process can be broken down into:

• Choosing a writing tool: 2 choices (pencil or pen).
• Selecting a paper type: 2 choices (lined or unlined paper).
• Deciding how to use the paper: 2 choices (one side or both sides).

Each decision is separate from the others, meaning that the choice of writing tool doesn't affect the choice of paper type. This separation is crucial for applying combinatorial methods.

Being clear about each decision point ensures accuracy when calculating combinations, as it avoids mistakenly omitting or duplicating options. This systematic approach is valuable in breaking down complex problems into manageable parts.

Combination calculation

Combination calculation involves multiplying the number of options for each independent choice to find the total number of combinations available. This strategy allows us to systematically determine all possible outcomes.

In Darius's scenario, the process is as follows:

• Identify the number of choices for each decision.
• Write down these numbers.
• Multiply them together to find the total combinations.

The math is straightforward: \( 2 \times 2 \times 2 = 8 \), which tells us there are 8 different ways Darius can do his homework.

This method is widely used in problems involving multiple steps or layers of decision-making, making it a vital tool in both educational settings and real-life situation assessments. It allows for structured planning and complete enumeration of possibilities, minimizing oversight.