Domanda del libro di testo
Use the graphs of the arithmetic sequences {a} and {b} to solve Exercises 51-58. If {an} is a finite sequence whose last term is -83, how many terms does {an} contain?
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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 53
Use the formula for nCr to solve Exercises 49–56. You volunteer to help drive children at a charity event to the zoo, but you can fit only 8 of the 17 children present in your van. How many different groups of 8 children can you drive?
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Identify the problem as a combination problem because the order in which the children are chosen does not matter. We want to find the number of ways to choose 8 children out of 17.
Recall the formula for combinations, which is given by \(\displaystyle nCr = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items, \(r\) is the number of items to choose, and \(!\) denotes factorial.
Substitute the given values into the formula: \(n = 17\) and \(r = 8\), so the expression becomes \(\displaystyle \binom{17}{8} = \frac{17!}{8!(17-8)!} = \frac{17!}{8!9!}\).
Calculate the factorial values or simplify the expression by expanding the factorials partially to make the calculation easier, for example, expand \$17!$ down to \$9!$ to cancel terms.
Evaluate the simplified expression to find the total number of different groups of 8 children that can be formed from 17 children.
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Combination Formula (nCr)
The combination formula, denoted as nCr, calculates the number of ways to choose r items from a set of n distinct items without regard to order. It is given by nCr = n! / [r! (n - r)!], where '!' denotes factorial. This formula is essential for counting groups or subsets where order does not matter.
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5:22
Combinations
Factorials
A factorial, represented 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 used in permutations and combinations to calculate the total number of arrangements or selections.
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5:22Factorials
Application of Combinations in Real-Life Problems
Combinations are used to determine how many different groups or selections can be made from a larger set when order does not matter. In this problem, selecting 8 children out of 17 to fit in a van is a practical example where combinations help find the number of possible groups.
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7:24Multiplying & Dividing Functions
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