Question

You own 4 pairs of jeans, 12 clean T-shirts, and 4 wearable pairs of sneakers. How many outfits (jeans, T-shirt, and sneakers) can you create?

Short answer

Answer: There are 192 possible outfits that can be created using the given items.

Step-by-step solution

Identify the number of ways to choose each item

There are 4 pairs of jeans, 12 T-shirts, and 4 pairs of sneakers. Let's denote the number of ways to choose jeans as J, the number of ways to choose T-shirts as T, and the number of ways to choose sneakers as S. So, J = 4, T = 12, and S = 4.

Use the counting principle to find the total number of possible outfits

According to the counting principle, the total number of possible outfits is the product of the number of ways to choose each item. So, the total number of outfits is given by the formula: Number of Outfits = J * T * S

Calculate the number of outfits

Substitute the values of J, T, and S from step 1 into the formula: Number of Outfits = 4 (jeans) * 12 (T-shirts) * 4 (sneakers) = 192 So, there are 192 possible outfits that can be created using the given items.

Key concepts

Counting Principle

The Counting Principle is a fundamental concept in probability and combinatorics. It helps in determining the number of possible combinations of different events. Essentially, if you can choose one item from one group in 'a' ways, another item from a second group in 'b' ways, and so on, then you can find the total number of combinations by multiplying these separate choices.
In our example, we have 4 pairs of jeans, 12 T-shirts, and 4 pairs of sneakers. By applying the Counting Principle, the total number of different outfits we can create is 192, calculated by multiplying the number of options for each clothing item: 4 * 12 * 4. This principle makes it easy to calculate all possible combinations by breaking down complex problems into simpler steps.
Understanding the Counting Principle is critical because it provides a structured approach to solving combinatorial problems efficiently. Instead of trying to list all combinations individually, this principle allows us to quickly and systematically determine the final count.

Combinatorics

Combinatorics is the branch of mathematics that studies counting, arrangement, and combination of objects. It is particularly concerned with the number of configurations that a set of objects can take. In this context, combinatorics makes use of formulas and principles like permutations and combinations to find answers to problems like how many outfits can be formed.
In solving our exercise, combinatorics are applied when using the Counting Principle to calculate the total number of outfit configurations. This involves multiplying the choices of jeans, T-shirts, and sneakers to find how many different sets can be assembled.

• It deals with problems of counting both permutations, where the order matters, and combinations, where the order does not matter.
• Combinatorics is essential in a wide range of applications from computer science to probability theory.

By understanding how to apply combinatoric techniques, learners can efficiently solve complex counting problems in various real-world situations.

Permutations

Permutations refer to arrangements of a set of items, where the order is important. In our scenario, we are concerned with different outfits, which suggests a different concept. Instead of needing permutations, we are dealing with simple combinations.
However, understanding permutations is valuable. For example, if we were interested in the different ways to arrange the shirts in a sequence, then we would look at permutations. The formula for permutations is \( n! \), where \( n \) is the number of items to arrange.
In problems where the order of items matters, permutations help us calculate how many ways we can arrange these items. While our specific problem focuses on combinations of choices from groups, permutations broaden the concept by adding the aspect of different arrangements within sequences.

Outfit Combinations

Outfit Combinations relate directly to the exercise question of how many different clothing sets we can put together. In this case, we have different categories of clothing items: jeans, T-shirts, and sneakers.
Each choice is independent, and selecting from each type forms a combination of one complete outfit. With 4 pairs of jeans, 12 T-shirts, and 4 sneakers, the combinations are calculated by multiplying the options available for each item: 4 * 12 * 4, resulting in a total of 192 different outfit combinations.

• The concept of outfit combinations exemplifies a real-world application of combinatorial mathematics.
• Such calculations can be used in various fields like commerce and fashion to determine stock possibilities or outfit planning.

By understanding how to calculate outfit combinations, we can appreciate the variety and flexibility in planning clothing alternatives, as well as apply mathematical concepts to everyday life scenarios.