How Many Outfit Combos With 3 Shirts & 2 Shorts?

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Hey guys! Ever wondered how many different outfits you can create with just a few items of clothing? Let's dive into a fun and practical math problem that you can totally relate to. Suppose our friend Kauã has 3 cool shirts and 2 awesome pairs of shorts. The big question is: How many different outfits can Kauã put together using these items? This is a classic example of a combinatorics problem, which is all about counting different possibilities. Stick around, and we'll break it down step by step so you can solve similar problems on your own!

Understanding the Basics of Combinations

First, let's make sure we're all on the same page. When we talk about combinations, we're talking about how many different ways we can select items from a larger set. In Kauã's case, he's selecting one shirt and one pair of shorts to create an outfit. To find the total number of combinations, we'll use a simple but powerful principle: the multiplication principle. This principle states that if you have 'm' ways to do one thing and 'n' ways to do another, then you have m * n ways to do both. Think of it like this: for each shirt Kauã chooses, he has multiple options for shorts. This multiplies the possibilities!

So, how does this apply to Kauã's wardrobe? He has 3 choices for shirts. Let's call them Shirt A, Shirt B, and Shirt C. For each of these shirts, he has 2 choices for shorts. Let's call them Shorts 1 and Shorts 2. If he picks Shirt A, he can pair it with either Shorts 1 or Shorts 2. That's two outfits right there! If he picks Shirt B, he can again pair it with Shorts 1 or Shorts 2, giving us two more outfits. And finally, if he picks Shirt C, he can still pair it with Shorts 1 or Shorts 2, adding another two outfits. To get the total number of outfits, we simply multiply the number of shirt choices by the number of shorts choices: 3 shirts * 2 shorts = 6 outfits. Easy peasy, right?

Step-by-Step Calculation: Shirts and Shorts

Let's break down the calculation in a more detailed, step-by-step manner to really nail down how we arrive at the answer. We'll start by listing out all the possible combinations to visualize exactly what's happening. This is super helpful, especially when you're first learning about combinations. It turns an abstract math problem into something very tangible.

  • Shirt A with Shorts 1: This is our first outfit. Cool and classic!
  • Shirt A with Shorts 2: Another great option with the same shirt but different shorts.
  • Shirt B with Shorts 1: Moving on to the second shirt, paired with the first pair of shorts.
  • Shirt B with Shorts 2: A different look using the second shirt.
  • Shirt C with Shorts 1: Our third shirt making its debut with the first pair of shorts.
  • Shirt C with Shorts 2: And finally, the third shirt paired with the second pair of shorts.

If you count them up, you'll see that we have a total of 6 different outfits. Now, let’s formalize this with our multiplication principle. Kauã has 3 options for shirts and, independently, 2 options for shorts. The total number of outfit combinations is the product of these options:

Total outfits = (Number of shirts) × (Number of shorts) = 3 × 2 = 6

This confirms our earlier calculation and shows how the multiplication principle works in practice. By understanding this principle, you can easily tackle similar problems with different numbers of items. For instance, what if Kauã had 4 shirts and 3 shorts? The same logic applies: 4 shirts × 3 shorts = 12 different outfits! The key is to identify the independent choices and multiply the number of options for each choice.

Visualizing Combinations: Tree Diagrams

Another fantastic way to visualize combinations is by using a tree diagram. Tree diagrams are graphical tools that help you map out all the possible outcomes in a step-by-step manner. They're particularly useful when you're dealing with a relatively small number of items, like in Kauã's case.

Here's how we can create a tree diagram for Kauã's outfit combinations:

  1. Start with the first choice: Kauã has 3 choices for shirts (Shirt A, Shirt B, Shirt C). Draw a single point at the beginning of your diagram. From this point, draw three branches, each representing one of the shirt choices. Label each branch with the corresponding shirt.
  2. Add the second choice: For each shirt choice, Kauã has 2 choices for shorts (Shorts 1, Shorts 2). At the end of each shirt branch, draw two more branches, each representing one of the shorts choices. Label each of these branches with the corresponding shorts.
  3. List the outcomes: Now, trace each path from the starting point to the end of the branches. Each path represents a unique outfit combination. You should have a total of 6 paths, corresponding to the 6 different outfits we calculated earlier.

By drawing the tree diagram, you can visually see all the possible combinations and confirm our mathematical calculation. This method is especially helpful for those who learn better visually. Plus, it can be used for more complex problems with multiple choices.

Real-World Applications of Combinations

The concept of combinations isn't just a math problem; it has numerous real-world applications. Understanding combinations can help you make better decisions in various aspects of your life. Here are a few examples:

  • Menu Planning: Imagine you're planning a meal and you have several options for appetizers, main courses, and desserts. By using combinations, you can figure out how many different meal combinations you can create.
  • Password Creation: When creating a password, you often have choices for the number of letters, numbers, and special characters. Understanding combinations can help you estimate the strength of your password.
  • Probability Calculations: Combinations are fundamental in probability calculations. For example, if you're trying to calculate the probability of winning a lottery, you need to understand how many different combinations of numbers are possible.
  • Event Planning: When organizing an event, you might have choices for the venue, catering, and entertainment. By using combinations, you can determine how many different event setups you can create.
  • Genetics: In genetics, combinations are used to calculate the possible genetic variations in offspring based on the genes of the parents.

By mastering the concept of combinations, you’ll not only excel in math class but also gain a valuable skill that can be applied in countless real-life situations.

Practice Problems to Sharpen Your Skills

Now that you understand the basics of combinations, it's time to put your knowledge to the test with some practice problems. Working through these exercises will help solidify your understanding and boost your confidence. Remember, the key is to identify the independent choices and apply the multiplication principle.

Problem 1:

Sarah has 5 different skirts and 4 different tops. How many different outfits can she create?

Solution:

Using the multiplication principle, we multiply the number of skirt choices by the number of top choices: 5 skirts × 4 tops = 20 outfits. Sarah can create 20 different outfits.

Problem 2:

John is packing for a trip. He has 2 pairs of pants, 3 shirts, and 2 pairs of shoes. How many different outfits can he make?

Solution:

Here, we have three independent choices: pants, shirts, and shoes. So, we multiply the number of options for each: 2 pants × 3 shirts × 2 shoes = 12 outfits. John can create 12 different outfits.

Problem 3:

A restaurant offers a lunch special where you can choose one appetizer, one main course, and one drink. There are 3 appetizers, 4 main courses, and 5 drinks to choose from. How many different lunch specials are possible?

Solution:

Again, we have independent choices for each part of the meal. So, we multiply the number of options for each: 3 appetizers × 4 main courses × 5 drinks = 60 lunch specials. There are 60 different lunch specials possible.

By working through these problems, you'll become more comfortable with the concept of combinations and be better prepared to tackle more complex scenarios. Keep practicing, and you'll be a combination master in no time!

Conclusion: Unleashing the Power of Combinations

So, to wrap it up, Kauã can create 6 different outfits with his 3 shirts and 2 pairs of shorts. We figured this out using the multiplication principle, which is a fundamental concept in combinatorics. But more than just getting the right answer, it's about understanding the process and how you can apply this to other problems. Whether you're figuring out meal combinations, planning outfits, or even estimating probabilities, the ability to understand and calculate combinations is a seriously useful skill.

Keep practicing, stay curious, and you'll find that math can be both fun and incredibly practical. Now go forth and conquer those combinations! You got this!