8P4 11C7
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![The expression in the image is a fraction involving permutations and combinations. It is written as:
\[
\frac{{^8P_4}}{{^{11}C_7}}
\]
### Explanation:
1. **Permutations (\(^nP_r\))**:
- \(^8P_4\) represents the number of ways to arrange 4 items out of 8 distinct items where the order matters. It is calculated using the formula:
\[
^nP_r = \frac{n!}{(n-r)!}
\]
So, \(^8P_4 = \frac{8!}{(8-4)!} = \frac{8!}{4!}\).
2. **Combinations (\(^nC_r\))**:
- \(^1C_7\) represents the number of ways to choose 7 items from 11 distinct items where the order does not matter. It is calculated using the formula:
\[
^nC_r = \frac{n!}{r!(n-r)!}
\]
So, \(^1C_7 = \frac{11!}{7! \cdot (11-7)!} = \frac{11!}{7! \cdot 4!}\).
This fraction represents the ratio of the number of permutations of 8 items taken 4 at a time to the number of combinations of 11 items taken 7 at a time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3116f481-fc06-46b9-bba6-a3c97987560e%2F731414c2-2730-4481-8aca-a0040f507b5e%2F0mhz2fc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The expression in the image is a fraction involving permutations and combinations. It is written as:
\[
\frac{{^8P_4}}{{^{11}C_7}}
\]
### Explanation:
1. **Permutations (\(^nP_r\))**:
- \(^8P_4\) represents the number of ways to arrange 4 items out of 8 distinct items where the order matters. It is calculated using the formula:
\[
^nP_r = \frac{n!}{(n-r)!}
\]
So, \(^8P_4 = \frac{8!}{(8-4)!} = \frac{8!}{4!}\).
2. **Combinations (\(^nC_r\))**:
- \(^1C_7\) represents the number of ways to choose 7 items from 11 distinct items where the order does not matter. It is calculated using the formula:
\[
^nC_r = \frac{n!}{r!(n-r)!}
\]
So, \(^1C_7 = \frac{11!}{7! \cdot (11-7)!} = \frac{11!}{7! \cdot 4!}\).
This fraction represents the ratio of the number of permutations of 8 items taken 4 at a time to the number of combinations of 11 items taken 7 at a time.
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