Problem 24 - Algebra and Trigonometry

Question

A woman has 5 blouses and 8 skirts How many different outfits can she wear?

Step-by-Step Solution

Answer

The woman can wear 40 different outfits.

Step 1 - Understand the Problem

Determine the total number of blouses and skirts the woman has. She owns 5 blouses and 8 skirts.

Step 2 - Concept of Outfits

An outfit consists of one blouse and one skirt.

Step 3 - Use Multiplication Principle

To find the total number of different outfits, multiply the number of choices for blouses by the number of choices for skirts: \[ 5 \text{ blouses} \times 8 \text{ skirts} \ = 40 \text{ different outfits} \]

Key Concepts

multiplication principle

The multiplication principle is a fundamental concept in combinatorics. It helps us calculate the total number of outcomes for a series of choices. If there are several stages in a decision, and for each stage, there are a specific number of choices, the total number of outcomes is the product of the choices for each stage. For example, if a woman has 5 choices of blouses and 8 choices of skirts, the multiplication principle tells us the total number of outfit combinations is: \( 5 \times 8 = 40 \). This principle can be applied to various situations where you need to count the number of possible combinations.

outfits combination

In the given exercise, the concept of outfits combination involves selecting one blouse and one skirt to form a complete outfit. Each outfit is a unique combination of one item from each category. To understand this better, let's break it down:

• One category is blouses, with 5 options.
• The other category is skirts, with 8 options.

By combining each blouse with each skirt, we explore every possible pairing. This means we match each blouse with all the skirts, leading to a total number of combinations. Here’s the math: \(5 \text{ blouses} \times 8 \text{ skirts} = 40 \text{ different outfits}\). So, the woman can wear 40 unique outfits.

counting methods

Counting methods in combinatorics are strategies used to count the number of ways we can arrange or select items. One of the simplest and most common methods is the multiplication principle, as used in this exercise. Here are some key points:

• **Basic Counting:** Simply list out all possible combinations (though this can get complicated with large numbers).
• **Pigeonhole Principle:** A way to prove that if items are put into containers, at least one container must hold more than one item if there are more items than containers.

In our case, we use the multiplication principle: for selecting an outfit, we multiply the number of blouses by the number of skirts. This straightforward method ensures that we accurately determine the total number of combinations possible. Using counting methods properly allows us to manage and solve combinatorial problems efficiently.