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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

Understand the Problem

Identify what we need to find: the total number of different outfits the woman can wear, given the number of blouses and skirts.
02

Identify the Items

List out the items: there are 5 blouses and 3 skirts.
03

Formulate the Combination Rule

To find the total number of outfits, multiply the number of blouses by the number of skirts: Total Outfits = (Number of Blouses) 脳 (Number of Skirts)
04

Perform the Multiplication

Perform the multiplication: 5 blouses 脳 3 skirts = 15 outfits.

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Key Concepts

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

counting principle
The counting principle is a fundamental concept in combinatorial mathematics. It helps us determine the total number of outcomes by breaking down a problem into smaller, more manageable parts.
For instance, consider our example where the woman has five blouses and three skirts. The first step is to calculate the number of ways she can choose each blouse independently.
This principle tells us that for each blouse, there are multiple choices (in this case, skirts) she can pair it with.
We simply need to multiply the number of choices at each step to find the total number of outcomes.
This leads us to the next core concept, the multiplication rule.
multiplication rule
The multiplication rule is directly tied to the counting principle and is a useful technique for finding the total number of combined outcomes. It states that if we have multiple stages in a process, we can find the total number of outcomes by multiplying the number of choices at each stage.
Let's break this down using our example:
- The woman can choose 1 out of 5 blouses.
- For each blouse she chooses, she has 3 skirt options.
According to the multiplication rule, we can determine the number of different outfits by multiplying the number of choices for each item. Thus, we calculate \[5 \times 3 = 15\] This means there are 15 possible blouse and skirt combinations or outfits.
outfits combination
This section is all about combining items from different categories to create unique combinations. In our problem, the woman needs to create outfits by combining blouses and skirts.
Here's how you can visualize it:
- Imagine each blouse as a separate choice.
- For each blouse, pair it with each of the skirts one by one.
You'd see that for each blouse, there are three unique combinations with the skirts.
This exercise demonstrates how combinatorial mathematics is practically applied in real-world scenarios.
Whenever you need to form a combination of items from different groups (like clothes, meals, etc.), think in terms of the counting principle and multiplication rule.
basic probability
Basic probability can be closely related to concepts from combinatorial mathematics. Probability helps us understand the likelihood of a particular outcome happening. While this problem doesn't directly deal with probability, understanding combinations and counting principles is foundational to grasping basic probability.
To tie it back to our example, if you randomly select an outfit, each combination (outfit) has \[\frac{1}{15}\] probability of being chosen, because there are 15 total outfits.
In a more complex scenario where each piece might have different probabilities, these foundational counting principles help us understand and calculate those probabilities.

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Most popular questions from this chapter

The weather forecast says there is a \(10 \%\) chance of rain on Thursday. Jim wakes up on Thursday and sees overcast skies. Since it has rained for the past three days, he believes that the chance of rain is more likely \(60 \%\) or higher. What method of probability assignment did Jim use?

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You are dealt 5 cards from a standard 52-card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering the following questions: (a) How many ways can 5 cards be selected from a 52-card deck? (b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck? (c) The remaining 2 cards must be different from the 3 chosen and different from each other. For example, if we drew three kings, the 4th card cannot be a king. After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. If we have three kings, then we can choose twos, threes, and so on. Of the 12 ranks remaining, we choose 2 of them and then select one of the 4 cards in each of the two chosen ranks. How many ways can we select the remaining 2 cards? (d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not like?
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