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: 192 different outfits

Step-by-step solution

Count the number of choices for jeans

There are 4 pairs of jeans, so there are 4 choices for jeans.

Count the number of choices for T-shirts

There are 12 clean T-shirts, so there are 12 choices for T-shirts.

Count the number of choices for sneakers

There are 4 wearable pairs of sneakers, so there are 4 choices for sneakers.

Apply the counting principle

The counting principle states that we can multiply the number of choices for each item to find the total number of possible combinations. In this case, we have: Total_outfits = Number_of_jeans * Number_of_T-shirts * Number_of_sneakers

Calculate the result

Substitute the values from steps 1, 2, and 3 into the equation to find the total number of outfits: Total_outfits = 4 * 12 * 4 Total_outfits = 192 So there are 192 different outfits that can be created using the given jeans, T-shirts, and sneakers.

Key concepts

Combinations

When we talk about combinations in mathematics, we refer to the selection of items from a larger pool where the order doesn't matter. For example, if you were picking fruit from a basket for a fruit salad, it wouldn't matter if you grabbed an apple first or last; the result would still be a fruit salad. Similarly, when creating an outfit from a selection of jeans, T-shirts, and sneakers, the order in which you choose the items doesn't matter; what matters is which items you choose.

Mathematically, combinations are a basic concept of combinatorics and are denoted as \( C(n, k) \), which represents the number of ways to choose \( k \) elements from a set of \( n \) elements. However, in the context of the exercise, we are not concerned with combinations because the choice of jeans, T-shirts, and sneakers does create a specific combination that defines an outfit, and here, the order of selection is acknowledged implicitly.

Since the context implies a specific selection process, we're talking about distinct outcomes, which is where permutation and the counting principle come into play, not the traditional combination formula.

Probability and Statistics

While this exercise does not explicitly involve probability and statistics, understanding combinations and counting principles lays the foundation for more advanced topics within these fields. In probability, we often calculate the likelihood of a specific combination of events, such as drawing a hand from a deck of cards or getting a certain outcome from rolling dice.

In terms of statistics, combinations help us understand sample spaces or the set of all possible outcomes, which is crucial for predictions and data analysis. For instance, knowing the total number of possible outfits, as calculated in the exercise, might help a clothing retailer forecast sales or manage inventory. While the calculation of the probability would require additional context and data, the counting techniques are integral in establishing the total number of outcomes from which any probability would be calculated.

Basic Counting Techniques

Basic counting techniques are essential tools in understanding how to approach problems involving quantities and possibilities. The most fundamental of these is the counting principle, also known as the fundamental principle of counting or the rule of product. This principle explains how to find the number of possible outcomes when there are multiple choices to be made, and each choice is independent of the others.

In the provided exercise, we see the counting principle at work. It simply states that if one event can occur in \(m\) ways and a second independent event can occur in \(n\) ways, then the total number of ways the two events can occur is \(m \times n\). The same logic applies to multiple events. Thus, we can find the total number of unique outfits by multiplying the number of options for jeans, T-shirts, and sneakers together. This technique simplifies the process of determining the number of possible combinations of items and is a foundational concept for more complex statistical calculations and for understanding probabilities.