How many ways to realize a set of population?

[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!

 

[teddd]

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