Problem 14 - Chemistry: An Atoms-Focused Approach

Question

Imagine you have four identical chairs to arrange on four steps leading up to a stage, one chair on each step. The chairs have numbers on their backs: \(1,2,3,\) and \(4 .\) How many different micro states for the chairs are possible? (When viewed from the front, all the micro states look the same. When viewed from the back, you can identify the different micro states because you can distinguish the chairs by their numbers.)

Step-by-Step Solution

Answer

Answer: 24 different microstates are possible.

Step 1: Understand the problem

The problem is asking us to find the number of unique ways to arrange the numbered chairs on the steps. There are four chairs and four steps, and we must place one chair on each step.

Step 2: Identify the concept of permutations

A permutation is an arrangement of objects in a specific order. In this case, we need to find the number of unique permutations of chairs (numbered \(1,2,3,\) and \(4\)) on the steps.

Step 3: Calculate the total number of permutations

We can calculate the total number of permutations using the formula: Total permutations = n! (where n refers to the total number of objects to arrange) Here, we have 4 chairs, so the total permutations will be: Total permutations = 4!

Step 4: Calculate the value of 4!

Now, we need to calculate the value of 4! (factorial of 4). The factorial of a number is the product of all positive integers less than or equal to that number. The factorial of 4 is: 4! = 4 × 3 × 2 × 1 = 24

Step 5: Interpret the result

The total number of microstates (unique permutations) for the chairs is 24. Therefore, there are 24 different ways to arrange the chairs with numbers \(1,2,3,\) and \(4\) on the steps.

Key Concepts

Factorial

When you're solving problems involving permutations, like arranging chairs on steps, understanding the concept of factorial is key. The factorial of a number, denoted as \( n! \), represents the multiplication of all positive integers up to that number. For example:

• \( 4! \) means multiplying 4, 3, 2, and 1 together, which equals 24.
• \( 5! \) would be 5 * 4 * 3 * 2 * 1, equaling 120.

Factoring a number provides a way to determine how many unique sequences can be formed with different items, each taken one at a time. It simplifies calculations in permutations because it instantly gives you the total number of possible arrangements. Always remember, the larger the number, the greater the factorial, which means more possible arrangements.

Arrangement

In any permutation problem, arrangement is about the order in which you set items. Unlike combinations where order doesn't matter, permutations care a lot about the sequence. For instance, arranging chairs numbered \(1, 2, 3, 4\) on different steps is a clear permutation task. Let's break it down:

• The position of each chair is important. For example, placing chair 1 on the first step and chair 2 on the second step is different from swapping their spots.
• Since all chairs are distinct in number, each unique order counts as a different arrangement.

When you calculate arrangements, you're essentially figuring out how many ways you can line up your items in varying sequences. Each unique sequence or order is a different permutation. This is why knowing how to determine arrangements is essential for permutation problems.

Combinatorics

Combinatorics is a vast field in mathematics that deals with counting, arrangement, and combination of objects. It plays a significant role when you're figuring out permutations, like how you numbered chairs are ordered on steps.In the context of arranging the chairs, combinatorics helps us answer questions like:

• How many ways can we arrange 'n' distinct objects? (For chairs, this means using factorial \( n! \)).
• What are the possible orders for placing them?

Combinatorics not only tells you how to count and arrange but also introduces you to more advanced concepts like probability and graph theory. Here, it fundamentally teaches you how different arrangements can be formed, which is crucial for solving problems with unique permutations. This builds an important foundation for understanding more complex mathematical concepts.