Problem 43
Question
A wholesale T-shirt company offers sizes small, medium, large, and extra-large in organic or non-organic cotton and colors white, black, gray, blue, and red. How many different Tshirts are there to choose from?
Step-by-Step Solution
Verified Answer
There are 40 different T-shirt combinations to choose from.
1Step 1: Determine the Number of Sizes
The T-shirt company offers the following sizes: small, medium, large, and extra-large. Therefore, there are 4 different sizes to choose from.
2Step 2: Determine the Number of Cotton Types
The T-shirt company offers shirts in two types of cotton: organic and non-organic. Thus, there are 2 cotton types available.
3Step 3: Determine the Number of Colors
The T-shirt company offers T-shirts in five colors: white, black, gray, blue, and red. Therefore, there are 5 different colors available.
4Step 4: Calculate the Total Number of Combinations
To find the total number of different T-shirts, multiply the number of options for each attribute together: the number of sizes, the number of cotton types, and the number of colors.\[\text{Total combinations} = 4 \times 2 \times 5 = 40\]So, there are 40 different T-shirt combinations to choose from.
Key Concepts
Counting PrinciplesMultiplication PrincipleCombinationsDiscrete Mathematics
Counting Principles
Counting principles are fundamental rules used to determine the number of ways an event can occur. In simple terms, it's a way to systematically count the number of outcomes of different choices. Consider this rule as a guideline for breaking complex problems into smaller, more manageable parts.
Let's say you have multiple decision points, like choosing a T-shirt that involves selecting a size, a cotton type, and a color. By using counting principles, you can figure out how many unique T-shirt combinations are possible. Each choice point contributes to a part of the total combinations you can have. These principles help us to avoid double-counting or missing any possible outcomes.
Let's say you have multiple decision points, like choosing a T-shirt that involves selecting a size, a cotton type, and a color. By using counting principles, you can figure out how many unique T-shirt combinations are possible. Each choice point contributes to a part of the total combinations you can have. These principles help us to avoid double-counting or missing any possible outcomes.
Multiplication Principle
The multiplication principle, also known as the rule of product, is a key concept in combinatorics. It's a simple yet powerful way to count the number of total outcomes when you have multiple choices to make.
For example, suppose you have 4 T-shirt sizes, 2 types of cotton, and 5 color options. To find out how many different T-shirts are possible, we use the multiplication principle. Multiply the number of options for each choice:
For example, suppose you have 4 T-shirt sizes, 2 types of cotton, and 5 color options. To find out how many different T-shirts are possible, we use the multiplication principle. Multiply the number of options for each choice:
- Size options: 4
- Cotton type options: 2
- Color options: 5
Combinations
Combinations refer to the ways in which items can be selected from a group, where the order does not matter. In this context, it's important to understand that combinations are used to calculate how many different groups can be formed.
When calculating T-shirt combinations, think about each attribute (like size, cotton type, and color) as having its own set of possible choices, and you combine them to form a complete product choice. However, unlike permutations, combinations do not concern themselves with arranging these attributes in any specific order because each attribute is fixed to one choice, leading to our final count of 40 unique products.
When calculating T-shirt combinations, think about each attribute (like size, cotton type, and color) as having its own set of possible choices, and you combine them to form a complete product choice. However, unlike permutations, combinations do not concern themselves with arranging these attributes in any specific order because each attribute is fixed to one choice, leading to our final count of 40 unique products.
Discrete Mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. It involves mathematical reasoning and combinatorial analysis, key for understanding counting principles and the multiplication principle.
In the context of the T-shirt example, discrete mathematics helps break down a seemingly complex decision-making process—choosing between various T-shirt options—into manageable steps that utilize discrete choices. Each choice (size, cotton type, color) is a discrete event, and understanding discrete mathematics is what empowers us to handle and compute these choices efficiently.
In the context of the T-shirt example, discrete mathematics helps break down a seemingly complex decision-making process—choosing between various T-shirt options—into manageable steps that utilize discrete choices. Each choice (size, cotton type, color) is a discrete event, and understanding discrete mathematics is what empowers us to handle and compute these choices efficiently.
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Problem 43
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