Question

A classroom has two rows of eight seats. There are 14 students in the class. Five students always sit in the front row, and four always sit in the back row. In how many ways can the students be seated?

Short answer

There are 145,152,000 possible seating arrangements.

Step-by-step solution

Analyze the Problem

We want to determine how many ways the class of 14 students can be seated in two rows of 8 seats when there are specific conditions on where some of them must sit. Five students must always be in the front row, and four must always be in the back row.

Choose Students with Conditions

First, select 5 students from the 14 to always sit in the front row. We will use combinations for this because the order does not matter.The number of ways to choose 5 students from 14 is calculated using the combination formula: \[\binom{14}{5} = \frac{14!}{5!(14-5)!} = \frac{14!}{5! \, 9!} = 2002.\]

Assign Selected Students to Specific Rows

Next, choose 4 students out of the remaining 9 students to sit in the back row. Again, the order in the selection does not matter, so we use combinations:\[\binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4! \, 5!} = 126.\]

Seat the Chosen Students in Their Own Rows

Once we have selected which students will be in the front row and which will be in the back row, we need to find the number of ways to arrange them in their seats.Arrange the 5 chosen front-row students:There are 5! ways to arrange them: \[5! = 120.\]Arrange the 4 chosen back-row students:There are 4! ways to arrange them: \[4! = 24.\]

Calculate Total Arrangements

The next step is to multiply all these possibilities together to get the total number of seating arrangements:\[\binom{14}{5} \times \binom{9}{4} \times 5! \times 4! = 2002 \times 126 \times 120 \times 24.\]Calculating this gives us:\[2002 \times 126 \times 120 \times 24 = 145,152,000.\]

Summary

In summary, taking into account the constraints of seating preferences, there are 145,152,000 ways to seat the 14 students in these two rows.

Key concepts

Combinatorial Analysis

Combinatorial analysis is the branch of mathematics that deals with counting, arranging, and analyzing the structure of finite structures. It focuses on how different combinations or arrangements of a set can be formed and, subsequently, counting these possible formations. This is particularly useful in scenarios like seating arrangements, scheduling, or any situation that requires systematic counting under certain constraints.

In our classroom seating example, combinatorial analysis helps us understand how to systematically calculate the total arrangements by breaking down the problem into manageable parts. It allows us to compute the number of ways to choose students for the front and back rows first, and then arrange them in their respective seats. This systematic approach ensures that all constraints are considered, and no possibilities are overlooked.

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorial mathematics used to count the arrangements or selections of a set of objects.

• Combinations: Used when the order of selection does not matter. In combinations, we are interested in selecting a subset from a larger set, without regard to the sequence. For example, choosing 5 students for the front row from a group of 14 uses combinations because we're interested only in who makes it to the front row, not the order they are chosen. This is represented as \( \binom{14}{5} \).


• Permutations: Used when the order of arrangement is important. After selecting the students who must sit in the front or back row, permutations help us figure out the number of ways those students can be arranged in their specific seats. This is represented using factorial notation such as \( 5! \) for the front row arrangement and \( 4! \) for the back row.

By combining these two concepts efficiently, we solve complex problems involving multiple stages of choice and arrangement.

Problem-Solving in Mathematics

Problem-solving in mathematics involves finding structured and logical approaches to arriving at solutions for mathematical problems. This encompasses a clear understanding of the problem, breaking it into smaller parts, and applying appropriate mathematical techniques.

When tackling the seating arrangement problem:

• Understand the Problem: First, comprehend the conditions (fixed rows for some students), what needs to be done (find total arrangements), and what tools to use (combinations and permutations).


• Divide and Conquer: Divide the problem into logical steps: choosing students for rows and arranging them separately.


• Execute Systematically: Use mathematical formulas, such as those for combinations \( \binom{n}{r} \) and permutations \( n! \), applying them to each part of the problem correctly to ensure a comprehensive solution.

This approach fosters a deeper understanding of not just how to solve a problem, but why each step is necessary and how it contributes to the overall solution.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to deduce properties or find solutions based on given information and rules. It's a key skill in developing problem-solving techniques and understanding the principles behind mathematical operations.

In the context of the classroom seating problem, mathematical reasoning guides us through the decision-making process:

• Interpreting Constraints: Reasoning begins with interpreting the conditions laid out for the problem, like specific students needing to sit in specified rows.


• Applying Knowledge: By using known formulas for combinations and permutations, we can reason the number of possible selections and arrangements effectively. For instance, calculating \( \binom{9}{4} \) reflects understanding that order of choice does not matter for picking students to stay in the back row.


• Ensuring Comprehensive Coverage: By multiplying combinations and permutations sequentially, reasoning ensures every possibility is accounted for, affirming the correctness of the total count of seating arrangements.

Mathematical reasoning makes these processes clear, leading to accurate and efficient solutions.