Textbook Question
In how many ways can five airplanes line up for departure on a runway?
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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 81
In Exercises 81–85, use a calculator's factorial key to evaluate each expression.
Verified step by step guidance1
Recall the definition of factorial: for a positive integer n, n! = n × (n-1) × (n-2) × ... × 1.
Express the given expression 200!/198! in terms of factorials to see if it can be simplified before calculating.
Write 200! as 200 × 199 × 198! to factor out 198! from the numerator and denominator.
Cancel out the common factorial term 198! in the numerator and denominator, leaving you with 200 × 199.
Use the calculator to multiply 200 by 199 to find the value of the expression 200!/198!.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factorial Definition
The factorial of a positive integer n, denoted n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials grow very quickly as n increases.
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Factorials
Simplifying Factorial Expressions
When dividing factorials like 200!/198!, common terms cancel out. Since 200! = 200 × 199 × 198!, the expression simplifies to 200 × 199. This simplification avoids calculating large factorials directly.
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Using a Calculator's Factorial Function
Many calculators have a factorial key (!) to compute factorials of integers. For large numbers, direct factorial calculation may be impractical, so simplifying expressions before using the calculator is essential to avoid errors or overflow.
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Related Practice
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?
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Use the Binomial Theorem to expand and then simplify the result: (x2 +x+ 1)3.
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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?
Textbook Question
In the sequence 21,700, 23,172, 24,644, 26,116,... which term is 314,628?
Textbook Question
Find the term in the expansion of (x2 + y2)5 containing x4 as a factor.
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