Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo

Larson 8th Edition

Ch. 3 - Probability

Problema 3.4.25

Capitolo 3, Problema 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?

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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.

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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.

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Percorso guidato

07:11

Introduction to Permutations

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.

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Percorso guidato

07:11

Introduction to Permutations

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.

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Percorso guidato

04:04

Fundamental Counting Principle

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