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

1. 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.
2. 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).
3. Step 3: Substitute the values into the formula: P(21, 6) = 21! / (21 - 6)!. This simplifies to P(21, 6) = 21! / 15!.
4. 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.
5. 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.

Key Concepts

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.