In A Closet There Are 8 Different Colors T -shirts. A Student Wants To Choose 2 Shirts To Take A Trip. How Many Different Combinations Of T -shirts Can You Choose?

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Introduction

When it comes to choosing items from a collection, we often encounter situations where the order of selection doesn't matter. This is a classic example of a combination problem in mathematics. In this article, we will delve into the world of combinations and explore how to calculate the number of different combinations of t-shirts that can be chosen from a closet containing 8 different colors.

What are Combinations?

Combinations are a way to calculate the number of ways to choose items from a collection, where the order of selection does not matter. In other words, if we choose two items from a collection, the combination is the same regardless of the order in which they were chosen. For example, if we choose two t-shirts from a closet, the combination is the same whether we choose the red t-shirt first and then the blue t-shirt, or the blue t-shirt first and then the red t-shirt.

Calculating Combinations

To calculate the number of combinations of t-shirts that can be chosen from a closet containing 8 different colors, we can use the formula for combinations:

C(n, k) = n! / (k!(n-k)!)

where:

  • n is the total number of items in the collection (in this case, 8 t-shirts)
  • k is the number of items to be chosen (in this case, 2 t-shirts)
  • ! denotes the factorial function, which is the product of all positive integers up to that number (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120)

Applying the Formula

Using the formula for combinations, we can calculate the number of combinations of t-shirts that can be chosen from a closet containing 8 different colors:

C(8, 2) = 8! / (2!(8-2)!) = 8! / (2!6!) = (8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / ((2 × 1)(6 × 5 × 4 × 3 × 2 × 1)) = (40320) / ((2)(720)) = 40320 / 1440 = 28

Interpretation

The result of the calculation, 28, represents the number of different combinations of t-shirts that can be chosen from a closet containing 8 different colors. This means that there are 28 unique ways to choose 2 t-shirts from the closet, regardless of the order in which they were chosen.

Real-World Applications

Combinations have numerous real-world applications, including:

  • Team selection: In sports, teams are often selected from a pool of players. The number of combinations of players that can be chosen for a team is an example of a combination problem.
  • Menu planning: Restaurants often offer a variety of menu items. The number of combinations of menu items that can be chosen by a customer is an example of a combination problem.
  • Travel planning: When planning a trip, travelers often need to choose from a variety of activities, such as sightseeing, dining, and entertainment. The number of combinations of activities that can be chosen is an example of a combination problem.

Conclusion

In conclusion, combinations are a fundamental concept in mathematics that has numerous real-world applications. By understanding how to calculate combinations, we can solve a wide range of problems that involve choosing items from a collection, where the order of selection does not matter. In this article, we explored how to calculate the number of combinations of t-shirts that can be chosen from a closet containing 8 different colors, and we saw that the result is 28 unique combinations.

Additional Examples

To further illustrate the concept of combinations, let's consider a few additional examples:

  • Choosing 3 items from a collection of 5: Using the formula for combinations, we can calculate the number of combinations of 3 items that can be chosen from a collection of 5 items: C(5, 3) = 5! / (3!(5-3)!) = 5! / (3!2!) = (5 × 4 × 3 × 2 × 1) / ((3 × 2 × 1)(2 × 1)) = (120) / ((6)(2)) = 120 / 12 = 10

  • Choosing 4 items from a collection of 6: Using the formula for combinations, we can calculate the number of combinations of 4 items that can be chosen from a collection of 6 items: C(6, 4) = 6! / (4!(6-4)!) = 6! / (4!2!) = (6 × 5 × 4 × 3 × 2 × 1) / ((4 × 3 × 2 × 1)(2 × 1)) = (720) / ((24)(2)) = 720 / 48 = 15

Frequently Asked Questions

In this article, we will address some of the most common questions related to combinations.

Q: What is the difference between combinations and permutations?

A: Combinations and permutations are both used to calculate the number of ways to choose items from a collection, but they differ in the order of selection. Permutations take into account the order of selection, while combinations do not.

Q: How do I calculate the number of combinations of items that can be chosen from a collection?

A: To calculate the number of combinations of items that can be chosen from a collection, you can use the formula for combinations:

C(n, k) = n! / (k!(n-k)!)

where:

  • n is the total number of items in the collection
  • k is the number of items to be chosen
  • ! denotes the factorial function, which is the product of all positive integers up to that number

Q: What is the formula for combinations?

A: The formula for combinations is:

C(n, k) = n! / (k!(n-k)!)

Q: How do I calculate the number of combinations of 3 items that can be chosen from a collection of 5 items?

A: Using the formula for combinations, we can calculate the number of combinations of 3 items that can be chosen from a collection of 5 items:

C(5, 3) = 5! / (3!(5-3)!) = 5! / (3!2!) = (5 × 4 × 3 × 2 × 1) / ((3 × 2 × 1)(2 × 1)) = (120) / ((6)(2)) = 120 / 12 = 10

Q: How do I calculate the number of combinations of 4 items that can be chosen from a collection of 6 items?

A: Using the formula for combinations, we can calculate the number of combinations of 4 items that can be chosen from a collection of 6 items:

C(6, 4) = 6! / (4!(6-4)!) = 6! / (4!2!) = (6 × 5 × 4 × 3 × 2 × 1) / ((4 × 3 × 2 × 1)(2 × 1)) = (720) / ((24)(2)) = 720 / 48 = 15

Q: What are some real-world applications of combinations?

A: Combinations have numerous real-world applications, including:

  • Team selection
  • Menu planning
  • Travel planning
  • Product design
  • Marketing research

Q: How do I use combinations in real-world scenarios?

A: To use combinations in real-world scenarios, you can apply the formula for combinations to calculate the number of combinations of items that can be chosen from a collection. For example, if you are planning a trip and want to choose 3 activities from a list of 5 activities, you can use the formula for combinations to calculate the number of combinations of 3 activities that can be chosen from a collection of 5 activities.

Q: What are some common mistakes to avoid when using combinations?

A: Some common mistakes to avoid when using combinations include:

  • Not considering the order of selection
  • Not using the correct formula for combinations
  • Not calculating the number of combinations correctly
  • Not considering the total number of items in the collection

Q: How do I troubleshoot common issues with combinations?

A: To troubleshoot common issues with combinations, you can:

  • Check the formula for combinations to ensure that it is being used correctly
  • Verify that the total number of items in the collection is being used correctly
  • Recalculate the number of combinations to ensure that it is correct
  • Consider the order of selection to ensure that it is being taken into account correctly

Conclusion

In conclusion, combinations are a fundamental concept in mathematics that has numerous real-world applications. By understanding how to calculate combinations, you can solve a wide range of problems that involve choosing items from a collection, where the order of selection does not matter. In this article, we addressed some of the most common questions related to combinations and provided examples of how to use combinations in real-world scenarios.