Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 3 - Probability

Problem 3.4.25

Chapter 3, Problem 3.4.25

25. Playlist A band is preparing a setlist of 21 songs for a concert. How many different ways can the band play the first six songs?

Verified step by step guidance

Step 1: Recognize that the problem involves arranging a subset of songs (6 songs) from a larger set (21 songs). This is a permutation problem because the order in which the songs are played matters.

Step 2: Recall the formula for permutations, which is P(n, r) = n! / (n - r)!, where n is the total number of items (21 songs) and r is the number of items to arrange (6 songs).

Step 3: Substitute the values into the formula: P(21, 6) = 21! / (21 - 6)!. This simplifies to P(21, 6) = 21! / 15!.

Step 4: Simplify the factorial expression by canceling out the common terms in the numerator and denominator. Specifically, 21! = 21 × 20 × 19 × 18 × 17 × 16 × 15!, so the 15! in the numerator and denominator cancel out, leaving P(21, 6) = 21 × 20 × 19 × 18 × 17 × 16.

Step 5: Multiply the remaining terms (21 × 20 × 19 × 18 × 17 × 16) to find the total number of permutations. This will give the total number of ways the band can arrange the first six songs.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Permutations

Permutations refer to the different arrangements of a set of items where the order matters. In this context, since the band is selecting and arranging the first six songs from a total of 21, the number of permutations will determine how many unique sequences can be formed.

Factorial Notation

Factorial notation, denoted as n!, represents the product of all positive integers up to n. It is crucial for calculating permutations, as the number of ways to arrange k items from n is given by the formula n! / (n-k)!, where k is the number of items to arrange.

Combinatorial Counting

Combinatorial counting involves techniques for counting the arrangements or selections of items in a set. Understanding this concept helps in determining how many different ways the band can choose and order the first six songs from their setlist of 21.

Recommended video:

04:04

Fundamental Counting Principle

Related Practice

Textbook Question

Using a Pie Chart to Find Probabilities In Exercises 83-86, use the pie chart at the left, which shows the number of workers (in millions) by occupation for the United States. (Source: U.S. Bureau of Labor Statistics)

84. Find the probability that a worker chosen at random is not employed in a service occupation.

" style="" width="260">

126

views

Textbook Question

34. Lottery Number Selection A lottery has 52 numbers. In how many different ways can six of the numbers be selected? (Assume the order of selection is not important.)

126

views

Textbook Question

"Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.

13. Rolling a six-sided die and then rolling the die a second time so that the sum of the two rolls is five"

views

Textbook Question

True or False? In Exercises 3-6, determine whether the statement is true or false. If it is false,

explain why.

4. When two events are independent, they are also mutually exclusive.

150

views

Textbook Question

Graphical Analysis In Exercises 7 and 8, determine whether the events shown in the Venn diagram are mutually exclusive. Explain your reasoning.

130

views

Textbook Question

Identifying the Sample Space of a Probability Experiment In Exercises 25-32, identify the sample space of the probability experiment and determine the number of outcomes in the sample space. Draw a tree diagram when appropriate.

25. Guessing the initial of a student's middle name

160

views