Seating Arrangements with Specific Constraints: How Many Ways Can Three People Be Seated in Nine Seats Such That No Two Are Adjacent?

The problem of seating three people in nine seats arranged in a row with the constraint that no two people sit next to each other is a classic combinatorial problem that can be solved using principles of combinatorics. This article explores the different methods to solve this problem and explains the reasoning behind the solution.

Introduction to the Problem

We are tasked with seating three people in nine seats arranged in a row such that no two people sit next to each other. The identities of the people matter, meaning that each person is unique.

Initial Analysis and Solution

Let's start by considering the problem on a smaller scale. If we have five seats and three people, we can analyze the different ways to place the people without them sitting next to each other.

Step-by-Step Solution

1. **Initial Configuration with Five Seats:**

We start with five seats and three people. The number of ways to order three people is 3! 6. There is only one way to seat them in alternating positions (if we consider an alternating sequence).

2. **Adding the Sixth Seat:**

Relative to the alternating sequence, there are 4 places to add an extra seat. Therefore, there are 4 times 6 24 such seating arrangements.

Generalization to Six Seats

When we consider six seats (let's label them A, B, C, D, E, F) and need to seat persons 1, 2, and 3 such that no two sit next to each other, we have the following scenarios:

Scenario 1: Person 1 in seat A, Person 2 in seat C, Person 3 in seat E. This is just one arrangement, but there are 3! 6 ways to order the people. Scenario 2: Person 1 in seat B, Person 2 in seat D, Person 3 in seat F. This is just one arrangement, but there are 3! 6 ways to order the people. Scenario 3: Person 1 in seat A, Person 2 in seat C, Person 3 in seat F. This is just one arrangement, but there are 3! 6 ways to order the people. Scenario 4: Person 1 in seat B, Person 2 in seat D, Person 3 in seat E. This is just one arrangement, but there are 3! 6 ways to order the people. Scenario 5: Person 1 in seat A, Person 2 in seat B, Person 3 in seat E. This is just one arrangement, but there are 3! 6 ways to order the people. Scenario 6: Person 1 in seat A, Person 2 in seat C, Person 3 in seat D. This is just one arrangement, but there are 3! 6 ways to order the people.

The total number of arrangements is 6 times 6 36. However, we have double-counted some scenarios (e.g., A-B-C, B-C-A), so we need to correct for this. The correct number of unique arrangements is 4 times 6 24.

Using Combinatorial Principles

A more rigorous approach involves using combinatorial principles to determine the number of ways to select and arrange the seats.

We need to choose 3 seats such that no two are next to each other from a total of 9 seats. Imagining the seats from left to right, once the first chair is selected, we discard the chair immediately to the right. This leaves us with 4 available chairs for the next selection. After the second selection, we discard another chair, leaving 3 available chairs for the third selection. We choose 3 out of the 6 remaining chairs, which can be done in binom{4}{3} 4 ways. For each of these 4 ways, we can arrange the three people in 3! 6 ways. The total number of ways is 4 times 6 24.

Conclusion

The number of ways to seat three people in nine seats arranged in a row such that no two people sit next to each other is 24. This solution is derived using combinatorial principles and careful consideration of the constraints given.

Keywords: seating arrangements, combinatorics, adjacent seats.