***********... In how many ways can four people boarding a bus be seated if the bus has nine vacant seats? ways
***********... In how many ways can four people boarding a bus be seated if the bus has nine vacant seats? 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
![### Seating Arrangement Problem
**Problem Statement:**
In how many ways can four people boarding a bus be seated if the bus has nine vacant seats?
**Solution Approach:**
To determine the number of ways four people can be seated in nine vacant seats, you need to calculate the possible combinations and arrangements.
1. **Combinations:**
First, we need to choose 4 seats out of the 9 for the four people. This can be done using the combination formula \( \binom{n}{k} \), which is "n choose k".
\[
\binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4!5!}
\]
2. **Permutations:**
Once the 4 seats are chosen, we need to arrange 4 people in these 4 seats. This can be done in \( 4! \) (4 factorial) ways.
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]
3. **Total Number of Ways:**
Multiply the number of combinations by the number of permutations.
\[
\binom{9}{4} \times 4! = \frac{9!}{4!5!} \times 24
\]
Insert your answer into the box provided to find out the total number of ways four people can be seated in nine vacant seats.
**Interactive Help:**
If you need assistance with the calculations, click on the "Need Help?" button.
**Optional: Show My Work**
You can choose to show your detailed work by selecting the "Show My Work" option.
Understand this solution and solve similar problems with ease by doing these exercises on arrangement and combination.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e1b5fc2-7275-49e6-a434-e3b12d2d745f%2F716c754d-b751-4d88-baa1-f592f31f09a2%2Fnhphm08_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Seating Arrangement Problem
**Problem Statement:**
In how many ways can four people boarding a bus be seated if the bus has nine vacant seats?
**Solution Approach:**
To determine the number of ways four people can be seated in nine vacant seats, you need to calculate the possible combinations and arrangements.
1. **Combinations:**
First, we need to choose 4 seats out of the 9 for the four people. This can be done using the combination formula \( \binom{n}{k} \), which is "n choose k".
\[
\binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4!5!}
\]
2. **Permutations:**
Once the 4 seats are chosen, we need to arrange 4 people in these 4 seats. This can be done in \( 4! \) (4 factorial) ways.
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]
3. **Total Number of Ways:**
Multiply the number of combinations by the number of permutations.
\[
\binom{9}{4} \times 4! = \frac{9!}{4!5!} \times 24
\]
Insert your answer into the box provided to find out the total number of ways four people can be seated in nine vacant seats.
**Interactive Help:**
If you need assistance with the calculations, click on the "Need Help?" button.
**Optional: Show My Work**
You can choose to show your detailed work by selecting the "Show My Work" option.
Understand this solution and solve similar problems with ease by doing these exercises on arrangement and combination.
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