Domanda del libro di testo
Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence whose nth term is an = (3)5n Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 9, Problema 89
Evaluate n!/(n-r)! for n = 20 and r = 3
Guida verificata passo dopo passo1
Identify the given values: \( n = 20 \) and \( r = 3 \).
Recall the formula to evaluate: \( \frac{n!}{(n-r)!} \). This expression represents the number of permutations of \( n \) items taken \( r \) at a time.
Substitute the given values into the formula: \( \frac{20!}{(20-3)!} = \frac{20!}{17!} \).
Expand the factorial expressions to simplify the fraction. Since \( 20! = 20 \times 19 \times 18 \times 17! \), the \( 17! \) terms cancel out, leaving \( 20 \times 19 \times 18 \).
Multiply the remaining terms \( 20 \times 19 \times 18 \) to find the value of the expression.
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Factorial Notation
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. It is commonly used in permutations and combinations to count arrangements.
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Factorials
Permutation Formula
The expression n!/(n-r)! calculates the number of permutations of r objects chosen from n distinct objects. It counts the ordered arrangements and is essential for problems involving selection and ordering.
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7:11Introduction to Permutations
Substitution and Simplification
To evaluate n!/(n-r)!, substitute the given values for n and r, then simplify by canceling common factorial terms. This reduces the calculation to a product of consecutive integers, making it easier to compute.
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5:48Solving Systems of Equations - Substitution
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