Textbook Question
A club with 15 members is to choose four officers–president, vice president, secretary, and treasurer. In how many ways can these offices be filled?
Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 9, Problem 79
How many different ways can a director select 4 actors from a group of 20 actors to attend a workshop on performing in rock musicals?
Verified step by step guidance1
Recognize that this is a combination problem because the order in which the actors are selected does not matter.
The formula for combinations is given by: , where is the total number of items (actors) and is the number of items to choose (actors to select).
Substitute the values and into the formula: .
Simplify the denominator: , so the formula becomes: .
Cancel out the common terms in the numerator and denominator by expanding the factorials, leaving only the terms needed to compute the combination. This simplifies the calculation to: .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Combinations
Combinations refer to the selection of items from a larger set where the order of selection does not matter. In this context, the director is choosing 4 actors from a group of 20, which is a classic example of a combination problem. The formula for combinations is given by C(n, r) = n! / (r!(n - r)!), where n is the total number of items, r is the number of items to choose, and '!' denotes factorial.
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Combinations
Factorial
A factorial, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics, used to calculate the total arrangements or selections of items. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are essential in the combinations formula, as they help determine the number of ways to arrange or select items.
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Binomial Coefficient
The binomial coefficient, often represented as C(n, r) or 'n choose r', quantifies the number of ways to choose r elements from a set of n elements without regard to the order of selection. It is calculated using the formula C(n, r) = n! / (r!(n - r)!). This concept is crucial for solving problems involving selections, such as determining how many different groups of actors can be formed.
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