Calculating the Number of Possible Ways

Oct 13, 2025 by vinooalluri.com 73 views

Introduction

Mariah is a book lover who has a diverse collection of books across various genres. She has 5 mysteries, 7 biographies, and 8 science fiction novels that she wants to read. However, she can only choose three books to read from her collection. In this article, we will analyze the possible ways Mariah can choose three books from her collection and determine which of the given statements are true.

The Problem

Mariah has a total of 20 books in her collection, consisting of 5 mysteries, 7 biographies, and 8 science fiction novels. She wants to choose three books to read from her collection. The problem is to find the number of possible ways Mariah can choose three books from her collection.

Calculating the Number of Possible Ways

To calculate the number of possible ways Mariah can choose three books from her collection, we can use the combination formula. The combination formula is given by:

nCr = n! / (r!(n-r)!)

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

In this case, n = 20 (the total number of books) and r = 3 (the number of books to be chosen). Plugging these values into the combination formula, we get:

20C3 = 20! / (3!(20-3)!) = 20! / (3!17!) = (20 × 19 × 18) / (3 × 2 × 1) = 1140

Therefore, there are 1140 possible ways Mariah can choose three books from her collection.

Evaluating the Statements

Now that we have calculated the number of possible ways Mariah can choose three books from her collection, let's evaluate the given statements.

There are ${_{20}C_3}$ possible ways to choose three books. This statement is true, as we have calculated that there are indeed 1140 possible ways to choose three books from a collection of 20 books.
There are 1140 possible ways to choose three books. This statement is also true, as we have calculated that there are 1140 possible ways to choose three books from a collection of 20 books.
There are 20C3 possible ways to choose three books. This statement is true, as we have calculated that there are indeed 1140 possible ways to choose three books from a collection of 20 books.

Conclusion

In conclusion, Mariah has 1140 possible ways to choose three books from her collection of 20 books. The given statements are true, and we have evaluated each statement to determine its validity.

Mathematical Formulas Used

Combination formula: nCr = n! / (r!(n-r)!)
Factorial function: n! = n × (n-1) × (n-2) × ... × 2 × 1

References

"Combinatorics" by Richard P. Stanley
"Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
Mariah's Book Selection: A Mathematical Analysis - Q&A =====================================================

Introduction

In our previous article, we analyzed the possible ways Mariah can choose three books from her collection of 20 books. We calculated that there are 1140 possible ways to choose three books from a collection of 20 books. In this article, we will answer some frequently asked questions related to Mariah's book selection problem.

Q&A

Q: What is the combination formula used to calculate the number of possible ways to choose three books?

A: The combination formula is given by:

nCr = n! / (r!(n-r)!)

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

Q: How do we calculate the factorial function?

A: The factorial function is calculated by multiplying all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Q: What is the difference between permutation and combination?

A: Permutation refers to the arrangement of objects in a specific order, whereas combination refers to the selection of objects without considering the order.

Q: Can we use the permutation formula to calculate the number of possible ways to choose three books?

A: No, we cannot use the permutation formula to calculate the number of possible ways to choose three books. The permutation formula is used to calculate the number of arrangements of objects in a specific order, whereas the combination formula is used to calculate the number of selections of objects without considering the order.

Q: How do we calculate the number of possible ways to choose three books from a collection of 20 books using the combination formula?

A: To calculate the number of possible ways to choose three books from a collection of 20 books using the combination formula, we plug in the values n = 20 and r = 3 into the combination formula:

20C3 = 20! / (3!(20-3)!) = 20! / (3!17!) = (20 × 19 × 18) / (3 × 2 × 1) = 1140

Q: What is the significance of the number 1140 in the context of Mariah's book selection problem?

A: The number 1140 represents the number of possible ways Mariah can choose three books from her collection of 20 books.

Q: Can we use the combination formula to calculate the number of possible ways to choose three books from a collection of 25 books?

A: Yes, we can use the combination formula to calculate the number of possible ways to choose three books from a collection of 25 books. We plug in the values n = 25 and r = 3 into the combination formula:

25C3 = 25! / (3!(25-3)!) = 25! / (3!22!) = (25 × 24 × 23) / (3 × 2 × 1) = 2300

Q: What is the difference between the number of possible ways to choose three books from a collection of 20 books and a collection of 25 books?

A: The number of possible ways to choose three books from a collection of 20 books is 1140, whereas the number of possible ways to choose three books from a collection of 25 books is 2300.

Conclusion

In conclusion, we have answered some frequently asked questions related to Mariah's book selection problem. We have used the combination formula to calculate the number of possible ways to choose three books from a collection of 20 books and a collection of 25 books. We have also explained the significance of the number 1140 in the context of Mariah's book selection problem.

Mathematical Formulas Used

Combination formula: nCr = n! / (r!(n-r)!)
Factorial function: n! = n × (n-1) × (n-2) × ... × 2 × 1

References

"Combinatorics" by Richard P. Stanley
"Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer