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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

According to Internal Revenue Service records, \(6.42 \%\) of all household tax returns are audited. According to the Humane Society, \(39 \%\) of all households own a dog. Assuming dog ownership and audits are independent events, what is the probability a randomly selected household is audited and owns a dog?

A National Ambulatory Medical Care Survey administered by the Centers for Disease Control found that the probability a randomly selected patient visited the doctor for a blood pressure check is \(0.593 .\) The probability a randomly selected patient visited the doctor for urinalysis is 0.064. Can we compute the probability of randomly selecting a patient who visited the doctor for a blood pressure check or urinalysis by adding these probabilities? Why or why not?

For a parallel structure of identical components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.15 probability of failure. (a) Would it be unusual to observe one component fail? Two components? (b) What is the probability that a parallel structure with 2 identical components will succeed? (c) How many components would be needed in the structure so that the probability the system will succeed is greater than \(0.9999 ?\)

According to the U.S. National Center for Health Statistics, \(0.15 \%\) of deaths in the United States are 25 - to 34-year-olds whose cause of death is cancer. In addition, \(1.71 \%\) of all those who die are \(25-34\) years old. What is the probability that a randomly selected death is the result of cancer if the individual is known to have been \(25-34\) years old?

The following data represent the number of different communication activities (e.g., cell phone, text messaging, e-mail, Internet, and so on) used by a random sample of teenagers over the past 24 hours. $$ \begin{array}{lccccc} \text { Activities } & \mathbf{0} & \mathbf{1 - 2} & \mathbf{3 - 4} & \mathbf{5 +} & \text { Total } \\ \hline \text { Male } & 21 & 81 & 60 & 38 & \mathbf{2 0 0} \\ \hline \text { Female } & 21 & 52 & 56 & 71 & \mathbf{2 0 0} \\ \hline \text { Total } & \mathbf{4 2} & \mathbf{1 3 3} & \mathbf{1 1 6} & \mathbf{1 0 9} & \mathbf{4 0 0} \end{array} $$ (a) Are the events "male" and " 0 activities" independent? Justify your answer. (b) Are the events "female" and " \(5+\) activities" independent? Justify your answer. (c) Are the events \(" 1-2\) activities" and \(\cdot 3-4\) activities" mutually exclusive? Justify your answer. (d) Are the events "male" and "1-2 activities" mutually exclusive? Justify your answer.
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