Textbook Question
Solve: .
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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 84
Use a calculator's factorial key to evaluate each expression. 20!/(20−3)!
Verified step by step guidance1
Identify the expression given: \(\frac{20!}{(20-3)!}\).
Simplify the denominator inside the factorial: calculate \(20 - 3\) to get \(17\), so the expression becomes \(\frac{20!}{17!}\).
Recall the definition of factorial: \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\).
Express \$20!\( in terms of \)17!$ to simplify the fraction: \(20! = 20 \times 19 \times 18 \times 17!\).
Cancel out \$17!$ in numerator and denominator, leaving \(20 \times 19 \times 18\), which you can then multiply using a calculator.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factorials
A factorial, denoted by n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are commonly used in permutations, combinations, and probability calculations.
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Permutation Formula
The expression 20! / (20−3)! represents the number of permutations of 20 items taken 3 at a time. It counts the ordered arrangements of 3 elements selected from 20, calculated by dividing the factorial of the total items by the factorial of the difference.
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Using a Calculator's Factorial Function
Many scientific calculators have a factorial key (!) to compute factorial values quickly. To evaluate expressions like 20!/(20−3)!, you calculate 20! and 17! separately or use the permutation function if available, simplifying the process and reducing manual errors.
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Related Practice
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