
Question content area top

Part 1

A DJ is preparing a playlist of 

2121 

songs. How many different ways can the DJ arrange the first 

fivefive 

songs on the​ playlist?

Question content area bottom

Part 1

There are 

enter your response here 

different ways that the DJ can arrange the first 

fivefive 

songs on the playlist.

​(Type a whole​ number.)

To determine the number of ways the DJ can arrange the first five songs from a playlist of 2,121 songs, we need to calculate the number of possible permutations of five songs chosen from 2,121.

### Solution:
Since the order of songs matters in this context (it's an arrangement problem), we can use the **permutation formula**:

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

where:
- \( n \) is the total number of items (songs),
- \( r \) is the number of items to arrange (first 5 songs).

In this case:
- \( n = 2121 \)
- \( r = 5 \)

Substituting these values:

\[
P(2121, 5) = \frac{2121!}{(2121 - 5)!} = \frac{2121!}{2116!}
\]

### Calculation:
This simplifies to multiplying the first five descending values from 2,121:

\[
P(2121, 5) = 2121 \times 2120 \times 2119 \times 2118 \times 2117
\]

Calculating this product:

\[
P(2121, 5) = 2121 \times 2120 \times 2119 \times 2118 \times 2117 = 41844985532564
\]

Thus, there are **41,844,985,532,564** different ways the DJ can arrange the first five songs on the playlist.

Would you like more details or have any questions about this calculation?

---

Here are some related questions for further exploration:
1. How would the result change if the DJ wanted to arrange the first six songs instead?
2. What if the DJ only had 10 songs to choose from?
3. How many ways could the DJ arrange all 2,121 songs?
4. What if we consider combinations instead of permutations? How would that affect the result?
5. How does the formula for permutations differ from that for combinations?

**Tip:** Remember, permutations are used when the order of arrangement matters. For example, arranging songs in a specific order uses permutations, not combinations.

A DJ is preparing a playlist of 2,121 songs. How many different ways can the DJ arrange the first five songs on the playlist?

Explore the number of ways a DJ can arrange the first five songs from a playlist of 2,121. This problem involves permutation calculations using the formula P(n, r) = n! / (n - r)! and applies the Fundamental Counting Principle. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation