Domanda del libro di testo
In Exercises 49–52, a single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 2 the second time.
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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 51
Use the formula for nCr to solve Exercises 49–56. Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?
Guida verificata passo dopo passo1
Identify the problem as a combination problem where order does not matter. You want to find the number of ways to choose 4 books out of 12 without regard to order.
Recall the formula for combinations, which is given by \(nCr = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to choose.
Substitute the given values into the formula: \(n = 12\) and \(r = 4\), so the expression becomes \(\frac{12!}{4!(12-4)!}\).
Simplify the factorial expressions in the numerator and denominator to make the calculation easier. For example, expand \$12!$ as \(12 \times 11 \times 10 \times 9 \times 8!\) and cancel the \$8!$ in numerator and denominator.
Calculate the remaining multiplication and division to find the number of different collections of 4 books you can take.
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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 selections where order does not matter.
Video consigliato:
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
Counting Without Replacement
When selecting items from a set without replacement, each chosen item is not returned to the set, reducing the total number available for subsequent choices. Combinations apply here because the order of selection does not matter, only the group chosen.
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4:04Fundamental Counting Principle
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