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Chapter 5: Problem 32

A woman has five blouses and three skirts. Assuming that they all match, how many different outfits can she wear?

Short Answer

Expert verified
15 outfits

Step by step solution

01

- Identify the Number of Choices

Determine the number of blouses and the number of skirts available. Here, there are 5 blouses and 3 skirts.
02

- Understand the Concept of Combinations

Each blouse can be paired with each skirt. Therefore, the total number of outfits is the product of the number of choices for blouses and skirts.
03

- Apply the Multiplication Principle

Multiply the number of blouses by the number of skirts to determine the total number of outfits. \( 5 \times 3 = 15 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplication Principle
The Multiplication Principle, also known as the Counting Principle, is a key idea in combinatorics.
It states that if you have multiple sequential choices, the number of overall possibilities is the product of the number of choices at each step.
In this exercise, the woman has 5 choices for blouses and 3 choices for skirts.
  • First step: Choose a blouse (5 choices)
  • Second step: Choose a skirt that matches (3 choices)
By multiplying these together, we find that she can create 5 * 3 = 15 different outfits.
This principle is useful in many areas, including basic probability and everyday decision-making. It simplifies the calculation process when determining combinations of different items.
Basic Probability
Basic Probability helps us understand the likelihood of different outcomes occurring.
The probability of a specific event happening is calculated as the number of favorable outcomes divided by the total number of possible outcomes.
In our exercise, if the woman randomly picks an outfit, each blouse-skirt combination has an equal chance.
  • Number of favorable outcomes for any specific choice (e.g., a red blouse and black skirt): 1
  • Total number of possible outcomes: 15
Therefore, the probability of her picking any specific outfit is \(\frac{1}{15}\).
Basic Probability principles are often used to reason about outcomes and make predictions in various fields ranging from games to weather forecasting.
Fundamental Counting Principle
The Fundamental Counting Principle is essential for solving combinatorial problems.
It states that if there are p ways to do one thing and q ways to do another, you multiply these numbers to find the total ways both can be done.
In the given problem, the principle helps us determine the total outfits.

How it applies:
  • Number of ways to choose a blouse = 5
  • Number of ways to choose a skirt = 3
Applying the Fundamental Counting Principle: \(\text{Total outfits} = 5 \times 3 = 15\).
This concept extends beyond clothing combinations to tasks such as menu selections, travel routes, and organizational planning. It's a foundational idea that aids in efficient and accurate counting for complex decisions.

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Most popular questions from this chapter

A bag of 30 tulip bulbs purchased from a nursery contains 12 red tulip bulbs, 10 yellow tulip bulbs, and 8 purple tulip bulbs. (a) What is the probability that two randomly selected tulip bulbs are both red? (b) What is the probability that the first bulb selected is red and the second yellow? (c) What is the probability that the first bulb selected is yellow and the second is red? (d) What is the probability that one bulb is red and the other yellow?

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A Social Security number is used to identify each resident of the United States uniquely. The number is of the form \(x x x-x x-x x x x,\) where each \(x\) is a digit from 0 to 9 (a) How many Social Security numbers can be formed? (b) What is the probability of correctly guessing the Social Security number of the President of the United States?
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