Textbook Question
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.
1
views
Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
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 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?
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
4:04Fundamental Counting Principle
Related Practice
Textbook Question
The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. an = n + 5
Textbook Question
Use the Binomial Theorem to expand each expression and write the result in simplified form. (x1/3 +x-1/3)3
Textbook Question
Use the graphs of the arithmetic sequences {a} and {b} to solve Exercises 51-58.
Find a14+b12.
Textbook Question
Express each sum using summation notation. Use as the lower limit of summation and for the index of summation.
5
views
Textbook Question
Express each repeating decimal as a fraction in lowest terms. 0.6 (repeating 6)