Problem 24 Movie Selections The Foreign Lan... [FREE SOLUTION]

Chapter 4: Problem 24

Movie Selections The Foreign Language Club is showing a four-movie marathon of subtitled movies. How many ways can they choose 4 from the 11 available?

Short Answer

Expert verified

There are 330 ways to choose 4 movies from 11 available movies.

Step by step solution

Understanding the Problem

We are tasked with selecting 4 movies from a total of 11 available movies for a marathon. This selection process does not consider the order of selection, meaning it is a combination problem.

Using the Combination Formula

The number of ways to choose 4 movies from 11 is given by the combination formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]where \( n \) is the total number of movies (11) and \( r \) is the number of movies to choose (4).

Substituting Values

Substitute 11 for \( n \) and 4 for \( r \) in the combination formula: \[ C(11, 4) = \frac{11!}{4!(11-4)!} = \frac{11!}{4! \times 7!} \]

Calculating Factorials

Calculate the factorials:- \( 11! = 11 \times 10 \times 9 \times 8 \times 7! \) (We can ignore the rest of the terms since \( 7! \) will cancel out)- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

Simplifying the Expression

Substitute back into the expression:\[ C(11, 4) = \frac{11 \times 10 \times 9 \times 8 \times 7!}{4! \times 7!} \]The \( 7! \) terms cancel out, leaving:\[ C(11, 4) = \frac{11 \times 10 \times 9 \times 8}{24} \]

Solving the Expression

Perform the multiplication in the numerator:- \( 11 \times 10 = 110 \)- \( 110 \times 9 = 990 \)- \( 990 \times 8 = 7920 \)Now, divide by the denominator:\[ C(11, 4) = \frac{7920}{24} = 330 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Factorials

A fundamental concept in combinatorics is the use of factorials, often symbolized by the exclamation mark \( ! \). Understanding factorials is crucial when dealing with permutations and combinations. A factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \). It is denoted as \( n! \). For example:

\( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
\( 0! = 1 \) by definition, which is essential for simplifying equations

Factorials grow very rapidly as \( n \) increases. Therefore, in combination problems, you'll often see factorials that cancel out, simplifying the calculation, as we saw when canceling \( 7! \) in the denominator and numerator.

Using the Combination Formula

The combination formula is a tool for calculating the number of ways to choose items without regard to order, from a larger set. This is crucial in many mathematics problems, particularly those involving selection or arrangement.The formula is expressed as:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items to be chosen. In essence:

\( n! \) counts all possible arrangements of the entire set of items.
\( r! \) accounts for the arrangements of the items chosen.
\( (n-r)! \) handles the arrangements of the items not chosen.

This formula reflects the principle that when the order doesn鈥檛 matter, you must divide to remove repeated arrangements from the calculation.

Problem-Solving in Mathematics

Problem-solving in mathematics often involves understanding the core principles behind the formulas and applying them to real-world scenarios. With problems involving combinations, identify scenarios where order does not matter.Consider these steps when solving such problems:

**Identify the Total and the Selection:** Determine \( n \) and \( r \) for your problem. In the movie example, \( n = 11 \) and \( r = 4 \).
**Use the Correct Formula:** Apply the combination formula, given that order doesn't matter.
**Simplify the Calculation:** Use properties like the cancellation of similar factorials to ease computing large numbers.
**Calculate Carefully:** Ensure correct multiplication and division to avoid errors in your final answer.

These steps reflect a systematic approach, allowing you to focus on each part of the problem separately. This clarity cuts down on the chance of error and improves your confidence in mathematics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Movie Selections The Foreign Language Club is showing a four-movie marathon of subtitled movies. How many ways can they choose 4 from the 11 available?

Short Answer

Step by step solution

Understanding the Problem

Using the Combination Formula

Substituting Values

Calculating Factorials

Simplifying the Expression

Solving the Expression

Key Concepts

Understanding Factorials

Using the Combination Formula

Problem-Solving in Mathematics

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Decision Maths

Pure Maths

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.