Textbook Question
If you toss a fair coin six times, what is the probability of getting all heads?
Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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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 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?
Verified step by step guidance1
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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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 groups or subsets where order does not matter.
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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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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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Related Practice
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