Problem 8 - Algebra for College Students

Question

A medical researcher classifies subjects according to sex (female, male), smoking habits (smoker, nonsmoker), and weight (below average, average, above average). How many different combined classifications are used? 12

Step-by-Step Solution

Answer

There are 12 different combined classifications.

Step 1: Identify the Categories

The problem involves categorizing subjects based on three different features: 1. Sex, which can be either 'female' or 'male'. 2. Smoking habits, which can be either 'smoker' or 'nonsmoker'. 3. Weight, which can be 'below average', 'average', or 'above average'.

Step 2: Count the Options in Each Category

Each subject can be classified as: - **2** options for sex: female, male. - **2** options for smoking habits: smoker, nonsmoker. - **3** options for weight: below average, average, above average.

Step 3: Calculate the Total Number of Classifications

To find the total number of classifications, multiply the number of options in each category:\[2\text{ (sex)} \times 2\text{ (smoking habits)} \times 3\text{ (weight)} = 12\]

Step 4: Interpret the Result

The calculation shows that there are 12 different possible combinations when classifying subjects based on sex, smoking habits, and weight.

Key Concepts

Classification Problem

In combinatorics, a classification problem involves organizing objects into distinct groups based on certain criteria. In our case, the medical researcher is organizing subjects by their characteristics. Each characteristic the subjects can possess forms a category. Examples of such characteristics include sex, smoking habits, and weight.

The idea is that by identifying all possible traits or categories, we can classify each subject uniquely within each category. These traits define the different groups or combinations the subjects can belong to. Understanding classification problems helps in effective data management and analysis by organizing data logically and systematically.

In our exercise, identifying the specific categories is paramount to solving the classification problem. We know the attributes and the possible values for each attribute, guiding us to a solution by accurately defining the groupings.

Categorization

Categorization involves sorting or dividing objects into categories. It is a pivotal step in classification problems, as it sets boundaries and defines what each category contains. In our example involving medical subjects, categorization helps define each subject based on three attributes: sex, smoker status, and weight class.

Here's how each category is divided:

• **Sex:** Female or Male
• **Smoking Habits:** Smoker or Nonsmoker
• **Weight:** Below average, Average, or Above average

Being able to categorize each subject into one group for each attribute ensures that we can combine these categories to form comprehensive classifications. Each categorization aspect is mutually exclusive within its group, ensuring clarity and avoiding overlap of classifications.

By viewing data through the lens of categorization, we can simplify and streamline complex data, making it much easier to work with on a practical level.

Multiplication Principle

The multiplication principle is a fundamental concept in combinatorics used for counting the number of ways in which tasks can be performed. It states that if you have multiple stages or steps with a certain number of ways to complete each, the total number of ways to complete the entire series of steps is the product of the number of ways to complete each step.

In this exercise, we are determining the number of classifications by multiplying the options in each category:

• **Sex:** 2 options (female, male)
• **Smoking Habits:** 2 options (smoker, nonsmoker)
• **Weight:** 3 options (below average, average, above average)

Applying the multiplication principle, the total number of classifications is calculated as follows:\[2 \times 2 \times 3 = 12\]

This principle is powerful because it provides a simple way to compute complex probabilities and combinations efficiently. It allows us to manage diverse combinations seamlessly, facilitating problem-solving across many scientific and mathematical fields.