Select The Correct Answer.A School Bus Has 25 Seats, With 5 Rows Of 5 Seats. Fifteen Students From The First Grade And Five Students From The Second Grade Travel On The Bus. How Many Ways Can The Students Be Seated If All Of The Second-grade Students
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Introduction
In this article, we will delve into the world of combinatorics and explore the concept of seating arrangements. We will examine a specific scenario involving a school bus with 25 seats, divided into 5 rows of 5 seats. The bus is carrying 15 students from the first grade and 5 students from the second grade. Our goal is to determine the number of ways the students can be seated, given that all the second-grade students must be seated together.
Understanding the Problem
To tackle this problem, we need to break it down into manageable components. We have a total of 20 students, consisting of 15 first-graders and 5 second-graders. The second-graders must be seated together, which means we can treat them as a single entity or block. This block will occupy 5 seats, leaving 15 seats for the first-graders.
Calculating the Number of Ways to Seat the Second-Grade Students
Since the second-graders must be seated together, we can calculate the number of ways to arrange them within their block. There are 5 second-graders, and they can be arranged in 5! (5 factorial) ways. This is because each student can be seated in any of the 5 available seats, and there are 5 choices for the first seat, 4 choices for the second seat, and so on.
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
Calculating the Number of Ways to Seat the First-Grade Students
Now that we have seated the second-graders, we need to calculate the number of ways to seat the first-graders. There are 15 first-graders, and they can be seated in any of the 15 available seats. This is a permutation problem, as the order of seating matters. We can calculate the number of ways to seat the first-graders using the formula for permutations:
P(n, r) = \frac{n!}{(n-r)!}
In this case, n = 15 (the number of first-graders) and r = 15 (the number of seats available for the first-graders). Plugging in these values, we get:
P(15, 15) = \frac{15!}{(15-15)!} = \frac{15!}{0!} = 15!
Calculating the Total Number of Ways to Seat the Students
Now that we have calculated the number of ways to seat the second-graders and the first-graders, we can multiply these two values to get the total number of ways to seat the students. This is because each seating arrangement of the second-graders can be combined with each seating arrangement of the first-graders.
Total = 5! \times 15!
Evaluating the Expression
To evaluate the expression, we need to calculate the value of 15!. This is a large number, but we can use the formula for factorials to calculate it:
15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
Using a calculator or a computer program, we can evaluate this expression to get:
15! = 1,307,674,368,000
Now that we have calculated the value of 15!, we can multiply it by 5! to get the total number of ways to seat the students:
Total = 5! \times 15!
= 120 \times 1,307,674,368,000
= 156,894,417,600,000
Conclusion
In this article, we have explored the concept of seating arrangements and calculated the number of ways to seat 20 students on a school bus. We have treated the second-graders as a single block and calculated the number of ways to arrange them within their block. We have then calculated the number of ways to seat the first-graders and multiplied these two values to get the total number of ways to seat the students. The result is a staggering 156,894,417,600,000 possible seating arrangements.
Final Answer
The final answer is: 156,894,417,600,000
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Introduction
In our previous article, we explored the concept of seating arrangements and calculated the number of ways to seat 20 students on a school bus. We treated the second-graders as a single block and calculated the number of ways to arrange them within their block. We then calculated the number of ways to seat the first-graders and multiplied these two values to get the total number of ways to seat the students. In this article, we will answer some frequently asked questions related to seating arrangements.
Q&A
Q: What is the formula for calculating the number of ways to seat students?
A: The formula for calculating the number of ways to seat students is:
Total = 5! \times P(n, r)
Where 5! is the number of ways to arrange the second-graders within their block, and P(n, r) is the number of ways to seat the first-graders.
Q: What is the significance of treating the second-graders as a single block?
A: Treating the second-graders as a single block allows us to calculate the number of ways to arrange them within their block, which is 5!. This is because the second-graders must be seated together, and we can think of them as a single entity or block.
Q: How do we calculate the number of ways to seat the first-graders?
A: We calculate the number of ways to seat the first-graders using the formula for permutations:
P(n, r) = \frac{n!}{(n-r)!}
In this case, n = 15 (the number of first-graders) and r = 15 (the number of seats available for the first-graders).
Q: What is the total number of ways to seat the students?
A: The total number of ways to seat the students is:
Total = 5! \times 15!
This is because each seating arrangement of the second-graders can be combined with each seating arrangement of the first-graders.
Q: How do we evaluate the expression 15! ?
A: We evaluate the expression 15! by calculating the value of 15!:
15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
Using a calculator or a computer program, we can evaluate this expression to get:
15! = 1,307,674,368,000
Q: What is the final answer?
A: The final answer is:
Total = 5! \times 15!
= 120 \times 1,307,674,368,000
= 156,894,417,600,000
Conclusion
In this article, we have answered some frequently asked questions related to seating arrangements. We have explained the formula for calculating the number of ways to seat students, the significance of treating the second-graders as a single block, and how to calculate the number of ways to seat the first-graders. We have also evaluated the expression 15! and provided the final answer.
Final Answer
The final answer is: 156,894,417,600,000