Textbook Question
In Exercises 15-18, determine whether the situation involves permutations, combinations, or neither. Explain your reasoning.
15. The number of ways 16 floats can line up in a row for a parade
4. Probability
Counting
2:21 minutesProblem 3.4.25
Textbook Question
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 guidance1
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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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.
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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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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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