# How many ways to realize a set of population?

• teddd
In summary, the conversation discusses a problem involving a box containing 3 particles with different energy levels and how to calculate the number of ways the population of particles can be arranged. The person initially believes there are only two possibilities due to the distinguishability of the particles, but they later realize their mistake and calculate that there are actually three possible arrangements.
teddd
Hi everyone!

Here's my problem of the day:

Let's take a box containing 3 identical (but distinguishable) particles A B C. Let this be a canonical ensamble.

Suppose that A has energy $\varepsilon_0$ and both B and C have energy $\varepsilon_1$. We thereforre have 2 energy level, $n_0,n_1$. Take the number of states $g_{\alpha}$ in each energy level $\varepsilon_{\alpha}$ to be $1$.Now, I want to calculate in how many ways the set of population $\vec{n}=(n_0,n_1)$ can be realized.

At first sight I'd say that they're two: I can take $(A,BC)$ or $(A,CB)$, being the particle distinguishable.

But if I use the well-known Boltzmann forumula $$W(\vec{n})=N!\prod_{\alpha}\frac{ g_{\alpha}^{n_{\alpha}}}{n_{\alpha}}$$ and I put in the g's and n's I've taken above I get:$$W(\vec{n})=3! \left(\frac{1^1}{1!}\frac{1^2}{2!}\right)=3$$so there should be three ways to set up the vector $\vec{n}$!Where am I mistaking?? Thanks for help!

Last edited:
Ah, well, it's ok! I figured that out, I've done some serious rookie mistakes...

## 1. How can the number of ways to realize a set of population be calculated?

The number of ways to realize a set of population can be calculated by using the formula nPr = n! / (n-r)!, where n is the total number of items in the population and r is the number of items being selected from the population. This formula is known as the permutation formula and is commonly used in probability and combinatorics.

## 2. What is the difference between permutations and combinations?

Permutations and combinations are both ways of selecting items from a larger set, but they differ in their order and repetition. Permutations involve selecting items in a specific order, while combinations do not consider order. Additionally, permutations do not allow for repetition of items, whereas combinations do.

## 3. Can the number of ways to realize a set of population ever be infinite?

No, the number of ways to realize a set of population is always a finite number. This is because the total number of items in the population is finite, and therefore, the number of ways to select and arrange those items is also finite.

## 4. How can the concept of realizing a set of population be applied in real life?

The concept of realizing a set of population is commonly used in probability and statistics, such as in determining the likelihood of certain outcomes in a given situation. It can also be applied in fields such as genetics, where the number of possible gene combinations in a population can be calculated using permutation formulas.

## 5. Can the number of ways to realize a set of population change over time?

Yes, the number of ways to realize a set of population can change over time as the population itself changes. For example, if new items are added to the population or if the order of existing items is rearranged, the number of ways to realize the population will also change. However, if the total number of items remains the same, the number of ways to realize the population will remain constant.

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