Arranging Three People on Two Chairs: A Comprehensive Guide

The question of arranging three people on two chairs is a classic problem in combinatorics. This article will delve into the mathematical concepts involved, provide examples, and discuss the key principles underlying such arrangements.

Understanding Permutations and Combinations

Before diving into the problem, it's important to understand the concepts of permutations and combinations. A permutation is an arrangement of objects in a specific order, while a combination is a way of selecting objects from a set without regard to the order.

Applying Permutations to Our Problem

When arranging three people on two chairs, we need to find the number of possible permutations. This can be expressed using the formula:

Number of Permutations nPr n! / (n-r)!

Where:

n is the total number of items (in this case, people) r is the number of items to arrange (in this case, chairs) n! denotes the factorial of n, which is the product of all positive integers up to n.

Step-by-Step Calculation

Let's calculate the number of ways to arrange three people (labeled 1, 2, and 3) on two chairs using permutations:

Choose 2 people out of 3: This can be done in C(3, 2) ways.

C(3, 2) 3! / (2! * (3-2)!) 3! / (2! * 1!) 3

So, there are 3 ways to choose 2 people out of 3.

Arrange the 2 chosen people on the 2 chairs: This can be done in 2! ways.

2! 2 * 1 2

Combine the results: Multiply the number of ways to choose the people by the number of ways to arrange them.

Total arrangements C(3, 2) * 2! 3 * 2 6

Therefore, there are 6 ways to arrange three people on two chairs.

Example Arrangements

We can list all possible arrangements of three people (1, 2, and 3) on two chairs:

12, 21 13, 31 23, 32 12, 3 (This means person 3 stands) 3, 12 (This means person 3 sits while 1 and 2 stand) 13, 2 (This means person 2 stands) 2, 13 (This means person 2 sits while 1 and 3 stand) 23, 1 (This means person 1 stands) 1, 23 (This means person 1 sits while 2 and 3 stand) 3, 21 (This means person 1 sits while 2 and 3 stand) 23, 31 (This means person 1 sits while 2 and 3 stand)

These arrangements show that there are indeed 6 unique ways to sit two people and have one stand or sit separately, depending on the context.

Conclusion

In conclusion, the problem of arranging three people on two chairs involves permutations. By understanding and applying the concepts of combinations and permutations, we can determine that there are 6 unique ways to arrange three people on two chairs. This problem is not just a theoretical exercise but is also applicable in real-world scenarios, such as seating arrangements in small groups or seating designs in public spaces.