Problem 21 - Algebra 2 and Trigonometry

Question

Determine the number of possible outcomes. Assembling an outfit from 6 shirts, 4 pairs of pants, and 2 pairs of shoes

Step-by-Step Solution

Answer

There are 48 possible different outfits.

Step 1: Understanding the Problem

We need to determine the total number of possible outfits you can assemble if you have 6 different shirts, 4 different pairs of pants, and 2 different pairs of shoes. An outfit consists of one shirt, one pair of pants, and one pair of shoes.

Step 2: Using the Counting Principle

The Counting Principle states that if you have a certain number of choices for one event and another set of choices for a second event, you multiply the number of choices for each event to find the total number of possible combinations. Here, for each shirt, you can choose any of the 4 pairs of pants and any of the 2 pairs of shoes.

Step 3: Calculating Total Possible Combinations

According to the Counting Principle, the total number of different outfits is the product of the number of shirts, pants, and shoes. Therefore, the number of possible outcomes is calculated as follows: \[ ext{Total outcomes} = 6 imes 4 imes 2 \] This equals 48 different possible outfits.

Step 4: Conclusion

The total number of possible outcomes for assembling an outfit from the given options is 48. This means you can create 48 different combinations of outfits.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and combining objects. It is fundamental to understanding problems involving the arrangement of items. Combinatorial methods help solve questions like determining the number of possible outfits, as seen in the exercise, by considering the different ways items can be selected from a given set.
Combinatorics uses various approaches:

• Permutations: Arrangements where the order matters.
• Combinations: Selections where order does not matter.
• Counting Principles: Broad rules, such as the Multiplication Principle.

In the context of the original exercise, we use the counting principle to figure out how many ways we can choose one shirt, one pair of pants, and one pair of shoes among the given options. Understanding combinatorics is crucial for tackling problems related to arranging or selecting items in specific ways.

Probability

Probability measures how likely an event is to occur. It's expressed as a number between 0 and 1, where 0 indicates impossible and 1 means certain. Although the original exercise primarily involves counting combinations, knowing probability helps put these combinations in context, especially when trying to predict or analyze real-world scenarios.
For situations such as assembling an outfit:

• Calculate total possible outcomes — in our example, 48.
• Identify favorable outcomes — the desired result, like a specific outfit.
• Divide favorable outcomes by total outcomes to find probability.

While probability isn’t directly calculated in the exercise, understanding it can give perspective on how likely a specific result (like wearing a particular outfit) can occur compared to the entire set of possibilities.

Multiplication Principle of Counting

The Multiplication Principle, or Principle of Counting, is a key combinatorial tool used to figure out the total number of ways different choices can be combined. It states that if there are multiple stages or categories and each stage involves a choice, you multiply the number of ways each choice can be made to get the total number of possible outcomes.
In our outfit example:

• Shirts: 6 choices
• Pants: 4 choices
• Shoes: 2 choices

You multiply these numbers: \(6 \times 4 \times 2 = 48\) to find all possible combinations.
This principle is fundamental in both simple scenarios, like choosing an outfit, and more complex arrangements in mathematics and real-world applications, showcasing how individual decisions together create multiple complex outcomes.