Answered

Find the best answers to your questions with the help of IDNLearn.com's expert contributors. Get prompt and accurate answers to your questions from our community of experts who are always ready to help.

Select the correct answer.

A school bus has 25 seats, with 5 rows of 5 seats. 15 students from the first grade and 5 students from the second grade travel in the bus. How many ways can the students be seated if all of the second-grade students occupy the first row?

A. [tex][tex]${ }_{25} P _{20}$[/tex][/tex]
B. [tex][tex]${ }_5 P _5 \times{ }_{20} P _{15}$[/tex][/tex]
C. [tex][tex]${ }_5 C _5 \times{ }_{25} C _{14}$[/tex][/tex]
D. [tex][tex]${ }_5 P _5 \times{ }_{15} P _{15}$[/tex][/tex]
E. [tex][tex]${ }_5 P_5 \times{ }_{25} C _{15}$[/tex][/tex]

Sagot :

GinnyAnsweridn
To solve this problem, we need to find the number of ways the students can be seated given the constraints of the problem.

1. We have a school bus with 25 seats, arranged in 5 rows of 5 seats each.
2. There are 15 students from the first grade and 5 students from the second grade.
3. All the second-grade students must sit in the first row, which has exactly 5 seats.

### Step-by-Step Solution:

#### Step 1: Seating the Second-Grade Students
Since all 5 second-grade students must sit in the first row, we need to arrange these 5 students in 5 seats. The number of ways to arrange 5 students in 5 seats (a permutation) is calculated as [tex]\( 5P5 \)[/tex].

[tex]\[ 5P5 = \frac{5!}{(5-5)!} = 5! \][/tex]

#### Step 2: Seating the First-Grade Students
After seating the second-grade students, there are 20 seats left (since the first row is fully occupied). Now, we need to arrange 15 first-grade students in these 20 remaining seats. The number of ways to arrange 15 students in 20 seats (a permutation) is calculated as [tex]\( 20P15 \)[/tex].

[tex]\[ 20P15 = \frac{20!}{(20-15)!} = \frac{20!}{5!} \][/tex]

#### Step 3: Combining Both Arrangements
To find the total number of ways to seat all the students in the bus, we need to multiply the number of ways to arrange the second-grade students with the number of ways to arrange the first-grade students.

[tex]\[ 5P5 \times 20P15 = 5! \times \frac{20!}{5!} \][/tex]

Simplifying and calculating:

[tex]\[ 5P5 = 120 \][/tex]
[tex]\[ 20P15 = \frac{20!}{5!} = 2432902008176640000 \][/tex]

Therefore, the total number of arrangements is:

[tex]\[ 5P5 \times 20P15 = 120 \times 2432902008176640000 = 2432902008176640000 \][/tex]

#### Conclusion
The correct option is B:
[tex]\[ 5P5 \times 20P15 \][/tex]

So, the answer is option B: [tex]\( \boxed{5P5 \times 20P15} \)[/tex].
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is dedicated to providing accurate answers. Thank you for visiting, and see you next time for more solutions.