The question is about understanding how they define the energy levels in a derivation from Wikipedia. It is attached with a link to the internet site.
The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.
We have N objects and we need to choose objects to put in box 1.
There is no importance to the order, hence the number of possible choices are:
Now we are left with a total of particles and we need to choose from group objects to put in box b.
the number of possible configurations to this is:
We can continue this process for box k-1, and the number of ways to put objects in this box is:
And for box k:
And so forth and so on.
These are the number of ways we can arrange individual boxes.
Hence the number of ways we can distribute the objects in all the boxes is:
If we write it explicitly we see that factors in the numerator and denominator of successive terms cancel each other out:
However, if we distribute all the objects then
And we are left with:
This is the number of possible ways we can distribute N objects in k boxes. where box i can contain Ni objects.
Before we go on let's understand the idea behind degeneracy.
For that we look at a quantum one dimensional infinite potential well.
The one dimensional ...
The solution discusses the Boltzmann distribution problem about defining energy.