The General Manager Of A Fast-food Restaurant Chain Must Select 2 Restaurants From 5 For A Promotional Program. How Many Different Possible Ways Can This Selection Be Done?It Is Possible To Select The Two Restaurants In _____ Different Ways.

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The General Manager's Dilemma: Counting Possible Ways to Select Restaurants

As a general manager of a fast-food restaurant chain, selecting the right restaurants for a promotional program can be a daunting task. With five restaurants to choose from, the task of selecting two restaurants for the program can be overwhelming. In this article, we will explore the concept of combinations and permutations to determine the number of different possible ways the selection can be done.

Before we dive into the problem, let's understand the concepts of combinations and permutations. A combination is a selection of items from a larger set, where the order of selection does not matter. On the other hand, a permutation is a selection of items from a larger set, where the order of selection does matter.

The general manager needs to select 2 restaurants from 5 for a promotional program. We need to find the number of different possible ways this selection can be done.

Using Combinations

Since the order of selection does not matter, we can use the concept of combinations to solve this problem. The formula for combinations is:

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

where n is the total number of items, k is the number of items to be selected, and ! denotes the factorial function.

In this case, we have 5 restaurants (n = 5) and we need to select 2 restaurants (k = 2). Plugging these values into the formula, we get:

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

Using Permutations

Alternatively, we can use the concept of permutations to solve this problem. However, since the order of selection does not matter, we need to divide the number of permutations by the number of ways to arrange the selected items.

The formula for permutations is:

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

In this case, we have 5 restaurants (n = 5) and we need to select 2 restaurants (k = 2). Plugging these values into the formula, we get:

P(5, 2) = 5! / (5-2)! = 5! / 3! = (5 × 4 × 3 × 2 × 1) / (3 × 2 × 1) = 120 / 6 = 20

However, since the order of selection does not matter, we need to divide the number of permutations by the number of ways to arrange the selected items, which is 2!. Therefore, the number of combinations is:

P(5, 2) / 2! = 20 / 2 = 10

In conclusion, the general manager can select 2 restaurants from 5 for a promotional program in 10 different possible ways. This can be calculated using the concept of combinations or permutations, depending on the specific requirements of the problem.

The concept of combinations and permutations has numerous real-world applications. For example, in marketing, understanding the number of possible ways to select a target audience can help in creating effective marketing campaigns. In finance, understanding the number of possible ways to select a portfolio of stocks can help in making informed investment decisions.

In conclusion, the general manager's dilemma is a classic example of a problem that can be solved using the concept of combinations and permutations. By understanding these concepts, we can make informed decisions in a variety of real-world scenarios.

  • "Combinations and Permutations" by Math Is Fun
  • "Combinations and Permutations" by Khan Academy
  • "Introduction to Combinations and Permutations" by MIT OpenCourseWare
  • "Combinations and Permutations" by Wolfram MathWorld
    The General Manager's Dilemma: Q&A

In our previous article, we explored the concept of combinations and permutations to determine the number of different possible ways the general manager of a fast-food restaurant chain can select 2 restaurants from 5 for a promotional program. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the difference between combinations and permutations?

A: Combinations and permutations are both used to select items from a larger set, but they differ in the order of selection. Combinations are used when the order of selection does not matter, while permutations are used when the order of selection does matter.

Q: How do I calculate combinations?

A: The formula for combinations is:

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

where n is the total number of items, k is the number of items to be selected, and ! denotes the factorial function.

Q: How do I calculate permutations?

A: The formula for permutations is:

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

However, if the order of selection does not matter, you need to divide the number of permutations by the number of ways to arrange the selected items, which is k!.

Q: What is the formula for combinations and permutations in Excel?

A: In Excel, you can use the following formulas:

  • Combinations: =COMBIN(n, k)
  • Permutations: =PERMUT(n, k)

Q: Can I use combinations and permutations for more than two items?

A: Yes, you can use combinations and permutations for more than two items. For example, if you have 5 restaurants and you want to select 3 restaurants for a promotional program, you can use the following formulas:

  • Combinations: C(5, 3) = 5! / (3!(5-3)!) = 10
  • Permutations: P(5, 3) = 5! / (5-3)! = 60

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

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

  • Marketing: Understanding the number of possible ways to select a target audience can help in creating effective marketing campaigns.
  • Finance: Understanding the number of possible ways to select a portfolio of stocks can help in making informed investment decisions.
  • Sports: Understanding the number of possible ways to select a team can help in making informed decisions about player selection.

Q: Can I use combinations and permutations for non-numerical data?

A: Yes, you can use combinations and permutations for non-numerical data. For example, if you have a list of names and you want to select 3 names for a promotional program, you can use the following formulas:

  • Combinations: C(5, 3) = 5! / (3!(5-3)!) = 10
  • Permutations: P(5, 3) = 5! / (5-3)! = 60

In conclusion, combinations and permutations are powerful tools that can be used to solve a variety of problems. By understanding these concepts, you can make informed decisions in a variety of real-world scenarios.

  • "Combinations and Permutations" by Math Is Fun
  • "Combinations and Permutations" by Khan Academy
  • "Introduction to Combinations and Permutations" by MIT OpenCourseWare
  • "Combinations and Permutations" by Wolfram MathWorld
  • "Combinations and Permutations in Excel" by Microsoft
  • "Combinations and Permutations in R" by R-bloggers
  • "Combinations and Permutations in Python" by Real Python