Christina Is Randomly Choosing Three Movies To Take On Vacation From Nine Action Movies, Seven Science Fiction Movies, And Four Comedies. Which Statement Is True?A. The Probability That Christina Will Choose Three Comedies Can Be Expressed As

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Introduction

Christina is planning a vacation and wants to choose three movies to watch from a collection of action movies, science fiction movies, and comedies. The problem is to determine the probability that Christina will choose three comedies from the given collection of movies. In this article, we will explore the problem and determine the correct statement.

The Problem

Christina has nine action movies, seven science fiction movies, and four comedies to choose from. She wants to choose three movies for her vacation. The problem is to find the probability that Christina will choose three comedies from the given collection of movies.

Calculating the Probability

To calculate the probability that Christina will choose three comedies, we need to use the concept of combinations. The number of ways to choose three comedies from four comedies is given by the combination formula:

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

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

In this case, n = 4 (the number of comedies) and k = 3 (the number of movies to choose). Plugging in these values, we get:

C(4, 3) = 4! / (3!(4-3)!) = 4! / (3!1!) = (4 × 3 × 2 × 1) / ((3 × 2 × 1) × 1) = 4

So, there are 4 ways to choose three comedies from four comedies.

The Total Number of Ways to Choose Three Movies

To calculate the total number of ways to choose three movies from the given collection, we need to use the concept of combinations again. The total number of movies is 9 + 7 + 4 = 20.

The number of ways to choose three movies from 20 movies is given by:

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

So, there are 1140 ways to choose three movies from the given collection.

The Probability of Choosing Three Comedies

Now that we have the number of ways to choose three comedies and the total number of ways to choose three movies, we can calculate the probability of choosing three comedies.

The probability of choosing three comedies is given by:

P(3 comedies) = (Number of ways to choose 3 comedies) / (Total number of ways to choose 3 movies) = 4 / 1140 = 1 / 285

So, the probability that Christina will choose three comedies can be expressed as 1/285.

Conclusion

In this article, we explored the problem of choosing movies for vacation and determined the probability that Christina will choose three comedies. We used the concept of combinations to calculate the number of ways to choose three comedies and the total number of ways to choose three movies. The probability of choosing three comedies is 1/285.

References

Discussion

Introduction

In our previous article, we explored the problem of choosing movies for vacation and determined the probability that Christina will choose three comedies. In this article, we will answer some frequently asked questions about the problem.

Q: What is the total number of movies in the collection?

A: The total number of movies in the collection is 9 + 7 + 4 = 20.

Q: How many ways can Christina choose three movies from the collection?

A: There are 1140 ways to choose three movies from the collection.

Q: What is the probability that Christina will choose three comedies?

A: The probability that Christina will choose three comedies is 1/285.

Q: How many ways can Christina choose three comedies from the four comedies?

A: There are 4 ways to choose three comedies from the four comedies.

Q: What is the formula for calculating the number of combinations?

A: The formula for calculating the number of combinations is:

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

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

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 set, but they differ in the order of the items. Combinations calculate the number of ways to choose items without regard to order, while permutations calculate the number of ways to choose items with regard to order.

Q: Can you give an example of how to use the combination formula?

A: Let's say we want to choose 2 items from a set of 5 items. Using the combination formula, we get:

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

So, there are 10 ways to choose 2 items from a set of 5 items.

Q: What is the significance of the probability of choosing three comedies?

A: The probability of choosing three comedies is significant because it gives us an idea of the likelihood of a particular outcome. In this case, the probability of choosing three comedies is relatively low, which means that it is less likely to happen.

Conclusion

In this article, we answered some frequently asked questions about the problem of choosing movies for vacation. We hope that this article has provided you with a better understanding of the problem and the concepts involved.

References

Discussion

Do you have any other questions or comments about the article? Please feel free to share your thoughts in the discussion section below.