A pianist plans to play 5 pieces at a recital from her repertoire of 20 pieces, and is carefully considering which song to play first, second, etc. to create a good flow. How many different recital programs are possible?

MATLAB: An Introduction with Applications
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A pianist plans to play 5 pieces at a recital from her repertoire of 20 pieces, and is carefully considering which song to play first, second, etc. to create a good flow. How many different recital programs are possible?
**Problem Statement:**

A pianist plans to play 5 pieces at a recital from her repertoire of 20 pieces, and is carefully considering which song to play first, second, etc. to create a good flow. How many different recital programs are possible?

**Solution Explanation:**

To solve this problem, consider that the order in which the pieces are played matters. This situation involves permutations since you are arranging 5 pieces out of 20.

**Permutations Formula:**

The number of permutations of choosing \( r \) items from \( n \) items is given by the formula:

\[
P(n, r) = \frac{n!}{(n-r)!}
\]

**Application to the Problem:**

- \( n = 20 \) (total pieces)
- \( r = 5 \) (pieces to be played)

Using the formula:

\[
P(20, 5) = \frac{20!}{(20-5)!} = \frac{20!}{15!}
\]

Calculating this gives:

\[
= 20 \times 19 \times 18 \times 17 \times 16
\]

\[
= 1,860,480
\]

Therefore, there are **1,860,480** different possible recital programs.
Transcribed Image Text:**Problem Statement:** A pianist plans to play 5 pieces at a recital from her repertoire of 20 pieces, and is carefully considering which song to play first, second, etc. to create a good flow. How many different recital programs are possible? **Solution Explanation:** To solve this problem, consider that the order in which the pieces are played matters. This situation involves permutations since you are arranging 5 pieces out of 20. **Permutations Formula:** The number of permutations of choosing \( r \) items from \( n \) items is given by the formula: \[ P(n, r) = \frac{n!}{(n-r)!} \] **Application to the Problem:** - \( n = 20 \) (total pieces) - \( r = 5 \) (pieces to be played) Using the formula: \[ P(20, 5) = \frac{20!}{(20-5)!} = \frac{20!}{15!} \] Calculating this gives: \[ = 20 \times 19 \times 18 \times 17 \times 16 \] \[ = 1,860,480 \] Therefore, there are **1,860,480** different possible recital programs.
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