Three Out Of Seven Students In The Cafeteria Line Are Chosen To Answer Survey Questions. How Many Different Combinations Of Three Students Are Possible?$\[ {}_7 C_3=\frac{7!}{(7-3)!3!} \\]A. 210 B. 35 C. 7 D. 70

Feb 27, 2025 by ADMIN 217 views

Understanding Combinations: A Key Concept in Mathematics

When it comes to selecting a group of items from a larger set, understanding combinations is crucial. In this article, we will delve into the concept of combinations and explore how to calculate them. We will use a real-world example to illustrate the process and provide a step-by-step guide on how to arrive at the correct answer.

What are Combinations?

Combinations are a way to calculate the number of ways to select a group of items from a larger set, without regard to the order in which they are selected. This is in contrast to permutations, which take into account the order of selection. Combinations are commonly used in mathematics, statistics, and other fields to solve problems involving group selection.

The Formula for Combinations

The formula for calculating combinations is given by:

${ {}_n C_r = \frac{n!}{(n-r)!r!} }$

where:

${ n }$ is the total number of items in the set
${ r }$ is the number of items to be selected
${ n! }$ is the factorial of ${ n }$ , which is the product of all positive integers up to ${ n }$
${ (n-r)! }$ is the factorial of ${ n-r }$ , which is the product of all positive integers up to ${ n-r }$
${ r! }$ is the factorial of ${ r }$ , which is the product of all positive integers up to ${ r }$

A Real-World Example: Choosing Students to Answer Survey Questions

Let's consider a scenario where three out of seven students in a cafeteria line are chosen to answer survey questions. We want to find out how many different combinations of three students are possible.

Using the formula for combinations, we can plug in the values as follows:

${ {}_7 C_3 = \frac{7!}{(7-3)!3!} }$

To calculate the factorials, we can use the following values:

${ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 }$
${ (7-3)! = 4! = 4 \times 3 \times 2 \times 1 = 24 }$
${ 3! = 3 \times 2 \times 1 = 6 }$

Now, we can plug these values into the formula:

${ {}_7 C_3 = \frac{5040}{24 \times 6} }$

Simplifying the expression, we get:

${ {}_7 C_3 = \frac{5040}{144} }$

${ {}_7 C_3 = 35 }$

Therefore, there are 35 different combinations of three students possible.

Conclusion

In this article, we explored the concept of combinations and how to calculate them using the formula. We used a real-world example to illustrate the process and arrived at the correct answer. Combinations are a powerful tool in mathematics and statistics, and understanding how to calculate them is essential for solving problems involving group selection.

Key Takeaways

Combinations are a way to calculate the number of ways to select a group of items from a larger set, without regard to the order in which they are selected.
The formula for combinations is given by: ${ {}_n C_r = \frac{n!}{(n-r)!r!} }$
To calculate the factorials, we can use the following values: ${ n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1 }$
Using the formula, we can calculate the number of combinations of three students possible from a group of seven students.

Frequently Asked Questions

What is the difference between combinations and permutations?
How do I calculate the number of combinations of a group of items?
What is the formula for combinations?

Answer

Combinations and permutations are both used to calculate the number of ways to select a group of items from a larger set. However, combinations do not take into account the order of selection, while permutations do.
To calculate the number of combinations of a group of items, you can use the formula: ${ {}_n C_r = \frac{n!}{(n-r)!r!} }$
The formula for combinations is given by: ${ {}_n C_r = \frac{n!}{(n-r)!r!} }$
Combinations Q&A: Frequently Asked Questions and Answers

In our previous article, we explored the concept of combinations and how to calculate them using the formula. However, we understand that there may be more questions and concerns that you may have. In this article, we will address some of the frequently asked questions about combinations and provide answers to help you better understand this concept.

Q: What is the difference between combinations and permutations?

A: Combinations and permutations are both used to calculate the number of ways to select a group of items from a larger set. However, combinations do not take into account the order of selection, while permutations do. In other words, combinations are used to calculate the number of ways to select a group of items without regard to the order in which they are selected, while permutations are used to calculate the number of ways to select a group of items with regard to the order in which they are selected.

Q: How do I calculate the number of combinations of a group of items?

A: To calculate the number of combinations of a group of items, you can use the formula: ${ {}_n C_r = \frac{n!}{(n-r)!r!} }$ where:

${ n }$ is the total number of items in the set
${ r }$ is the number of items to be selected
${ n! }$ is the factorial of ${ n }$ , which is the product of all positive integers up to ${ n }$
${ (n-r)! }$ is the factorial of ${ n-r }$ , which is the product of all positive integers up to ${ n-r }$
${ r! }$ is the factorial of ${ r }$ , which is the product of all positive integers up to ${ r }$

Q: What is the formula for combinations?

A: The formula for combinations is given by: ${ {}_n C_r = \frac{n!}{(n-r)!r!} }$

Q: How do I calculate the factorials in the combination formula?

A: To calculate the factorials in the combination formula, you can use the following values:

${ n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1 }$
${ (n-r)! = (n-r) \times (n-r-1) \times (n-r-2) \times \ldots \times 2 \times 1 }$
${ r! = r \times (r-1) \times (r-2) \times \ldots \times 2 \times 1 }$

Q: Can I use a calculator to calculate combinations?

A: Yes, you can use a calculator to calculate combinations. Most calculators have a built-in function for calculating combinations, which can save you time and effort.

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

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

Statistics: Combinations are used to calculate the number of ways to select a sample from a population.
Computer Science: Combinations are used to calculate the number of ways to select a subset of items from a larger set.
Business: Combinations are used to calculate the number of ways to select a group of customers or employees.

Q: Can I use combinations to calculate permutations?

A: No, you cannot use combinations to calculate permutations. While combinations and permutations are related, they are not the same thing. Combinations are used to calculate the number of ways to select a group of items without regard to the order in which they are selected, while permutations are used to calculate the number of ways to select a group of items with regard to the order in which they are selected.

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

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

Not using the correct formula
Not calculating the factorials correctly
Not using the correct values for ${ n }$ and ${ r }$
Not checking the answer for errors

Conclusion

In this article, we addressed some of the frequently asked questions about combinations and provided answers to help you better understand this concept. We hope that this article has been helpful in clarifying any questions or concerns you may have had about combinations. If you have any further questions or concerns, please don't hesitate to ask.