Mariah Is Randomly Choosing Three Books To Read From The Following: 5 Mysteries, 7 Biographies, And 8 Science Fiction Novels. Which Of These Statements Are True? Choose Three Correct Answers.A. There Are \[${ }_5 C _3\$\] Possible Ways To
Introduction
Mariah is faced with a delightful problem: choosing three books to read from a diverse collection of 20 books, consisting of 5 mysteries, 7 biographies, and 8 science fiction novels. In this article, we will delve into the world of combinatorics to determine the number of possible ways Mariah can select her three books.
Combinations: The Key to Understanding
To solve this problem, we need to understand the concept of combinations. A combination is a selection of items from a larger set, where the order of selection does not matter. In this case, Mariah is selecting three books from a set of 20, and the order in which she selects them is irrelevant.
Calculating the Number of Combinations
The formula for calculating the number of combinations is given by:
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, Mariah has 20 books to choose from, and she wants to select 3 books. Plugging these values into the formula, we get:
C(20, 3) = 20! / (3!(20-3)!) = 20! / (3!17!) = (20 × 19 × 18) / (3 × 2 × 1) = 1140
Therefore, there are 1140 possible ways for Mariah to select three books from the collection of 20.
Analyzing the Statements
Now that we have calculated the total number of possible ways Mariah can select her three books, let's examine the statements provided:
A. There are {{ }_5 C _3$}$ possible ways to select 3 books from the 5 mysteries.
To calculate the number of possible ways to select 3 books from the 5 mysteries, we use the combination formula:
C(5, 3) = 5! / (3!(5-3)!) = 5! / (3!2!) = (5 × 4) / (2 × 1) = 10
Therefore, statement A is TRUE.
B. There are {{ }_7 C _3$}$ possible ways to select 3 books from the 7 biographies.
To calculate the number of possible ways to select 3 books from the 7 biographies, we use the combination formula:
C(7, 3) = 7! / (3!(7-3)!) = 7! / (3!4!) = (7 × 6 × 5) / (3 × 2 × 1) = 35
Therefore, statement B is TRUE.
C. There are {{ }_8 C _3$}$ possible ways to select 3 books from the 8 science fiction novels.
To calculate the number of possible ways to select 3 books from the 8 science fiction novels, we use the combination formula:
C(8, 3) = 8! / (3!(8-3)!) = 8! / (3!5!) = (8 × 7 × 6) / (3 × 2 × 1) = 56
Therefore, statement C is TRUE.
Conclusion
In conclusion, Mariah has 1140 possible ways to select three books from the collection of 20. The statements A, B, and C are all true, as they accurately represent the number of possible ways to select 3 books from the respective categories.
Final Thoughts
Q&A: Further Exploring Mariah's Book Selection Dilemma
Q: What is the total number of books Mariah has to choose from? A: Mariah has a total of 20 books to choose from, consisting of 5 mysteries, 7 biographies, and 8 science fiction novels.
Q: How many ways can Mariah select 3 books from the 5 mysteries? A: Mariah can select 3 books from the 5 mysteries in 10 ways, as calculated using the combination formula: C(5, 3) = 10.
Q: How many ways can Mariah select 3 books from the 7 biographies? A: Mariah can select 3 books from the 7 biographies in 35 ways, as calculated using the combination formula: C(7, 3) = 35.
Q: How many ways can Mariah select 3 books from the 8 science fiction novels? A: Mariah can select 3 books from the 8 science fiction novels in 56 ways, as calculated using the combination formula: C(8, 3) = 56.
Q: What is the total number of ways Mariah can select 3 books from the entire collection? A: Mariah can select 3 books from the entire collection in 1140 ways, as calculated using the combination formula: C(20, 3) = 1140.
Q: Can Mariah select the same book multiple times? A: No, Mariah cannot select the same book multiple times. The combination formula assumes that the order of selection does not matter, and that each book can only be selected once.
Q: Can Mariah select a book that is not in the collection? A: No, Mariah cannot select a book that is not in the collection. The combination formula only considers the books that are available in the collection.
Q: How can Mariah use this information to make her book selection? A: Mariah can use this information to make her book selection by considering the number of ways she can select 3 books from each category. She can then choose the category that interests her the most, or select a mix of books from different categories.
Q: What if Mariah wants to select 4 books instead of 3? A: If Mariah wants to select 4 books instead of 3, she can use the combination formula to calculate the number of ways she can do so. The formula would be: C(20, 4) = 4845.
Q: What if Mariah wants to select 2 books instead of 3? A: If Mariah wants to select 2 books instead of 3, she can use the combination formula to calculate the number of ways she can do so. The formula would be: C(20, 2) = 190.
Conclusion
In conclusion, Mariah's book selection dilemma is a great example of how combinatorics can be applied to real-world scenarios. By understanding the concept of combinations and using the formula to calculate the number of possible ways to select items, we can solve problems like Mariah's book selection dilemma.