K For a segment of a radio show, a disc jockey can play 5 records. If there are 8 records to select from, in how many ways can the program for this segment be arranged? ways ...
K For a segment of a radio show, a disc jockey can play 5 records. If there are 8 records to select from, in how many ways can the program for this segment be arranged? ways ...
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
22: Answer
![**Problem Statement**
For a segment of a radio show, a disc jockey can play 5 records. If there are 8 records to select from, in how many ways can the program for this segment be arranged?
**Explanation**
This problem involves calculating the number of permutations of 5 records selected from a total of 8 records. The order in which the records are played matters.
**Calculation Method**
The number of permutations of selecting 5 records from 8 can be calculated using the formula for permutations:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
Where:
- \( n \) is the total number of items to choose from (8 records in this case).
- \( r \) is the number of items to choose (5 records in this case).
Substituting into the formula:
\[ P(8, 5) = \frac{8!}{(8-5)!} = \frac{8!}{3!} \]
This calculates to:
\[ P(8, 5) = \frac{8 \times 7 \times 6 \times 5 \times 4}{1} = 6720 \]
**Conclusion**
Thus, there are 6,720 different ways the program can arrange and play the 5 records out of the 8 available.
**Note:** Be sure to fill in the calculation results in the space provided if this is for an interactive element on the website.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8470e22-cdb6-4814-babe-099f6875e962%2F4d8dc4f4-a775-4f2b-8ace-6bee1daf70c9%2Fy3sdifc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
For a segment of a radio show, a disc jockey can play 5 records. If there are 8 records to select from, in how many ways can the program for this segment be arranged?
**Explanation**
This problem involves calculating the number of permutations of 5 records selected from a total of 8 records. The order in which the records are played matters.
**Calculation Method**
The number of permutations of selecting 5 records from 8 can be calculated using the formula for permutations:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
Where:
- \( n \) is the total number of items to choose from (8 records in this case).
- \( r \) is the number of items to choose (5 records in this case).
Substituting into the formula:
\[ P(8, 5) = \frac{8!}{(8-5)!} = \frac{8!}{3!} \]
This calculates to:
\[ P(8, 5) = \frac{8 \times 7 \times 6 \times 5 \times 4}{1} = 6720 \]
**Conclusion**
Thus, there are 6,720 different ways the program can arrange and play the 5 records out of the 8 available.
**Note:** Be sure to fill in the calculation results in the space provided if this is for an interactive element on the website.
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