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Find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a60 when a1 = 35, d = -3.
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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 21
Evaluate each expression.
Guida verificata passo dopo passo1
Identify the components of the expression: \(\frac{^7P_3}{3!} - 7C_3\). Here, \(^7P_3\) represents the number of permutations of 7 items taken 3 at a time, \$3!$ is the factorial of 3, and \$7C_3$ is the number of combinations of 7 items taken 3 at a time.
Recall the formulas for permutations and combinations:
- Permutations: \(^nP_r = \frac{n!}{(n-r)!}\)
- Combinations: \(^nC_r = \frac{n!}{r!(n-r)!}\)
Calculate \(^7P_3\) using the permutation formula: \(^7P_3 = \frac{7!}{(7-3)!} = \frac{7!}{4!}\).
Calculate \$3!$ which is the factorial of 3: \(3! = 3 \times 2 \times 1\).
Calculate \$7C_3$ using the combination formula: \(7C_3 = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!}\).
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Permutation
A permutation refers to the arrangement of objects in a specific order. The notation nPr represents the number of ways to arrange r objects from a set of n distinct objects, calculated as n! / (n-r)!. Understanding permutations is essential when order matters in selection.
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Introduction to Permutations
Combination
A combination is a selection of objects where order does not matter. The notation nCr represents the number of ways to choose r objects from n distinct objects, calculated as n! / [r!(n-r)!]. Combinations are used when the arrangement order is irrelevant.
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5:22Combinations
Factorial and Simplification of Expressions
Factorials (n!) represent the product of all positive integers up to n and are fundamental in calculating permutations and combinations. Simplifying expressions involving factorials, permutations, and combinations requires careful manipulation to reduce complex terms and evaluate the expression correctly.
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5:22Factorials
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