I've been reading statistical mechanics, and I read the following on Wikipedia, on the article on Fermi-Dirac Statistics derivation in the micro-canonical ensemble :
Suppose we have a number of energy levels, labeled by index $i$, each level having energy $ε_i$ and containing a total of $n_i$ particles. Suppose each level contains $g_i$ distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta (i.e. their momenta may be along different directions), in which case they are distinguishable from each other, yet they can still have the same energy. The value of $g_i$ associated with level $i$ is called the "degeneracy" of that energy level. The Pauli exclusion principle states that only one fermion can occupy any such sublevel.
Then they have continued :
The number of ways of distributing $n_i$ indistinguishable particles among the $g_i$ sublevels of an energy level, with a maximum of one particle per sublevel is given by the binomial coefficient, using its combinatorial interpretation :
$$ w(n_{i},g_{i})={\frac {g_{i}!}{n_{i}!(g_{i}-n_{i})!}}$$ For example, distributing two particles in three sublevels will give population numbers of $110$, $101$, or $011$ for a total of three ways which equals $\frac{3!}{(2!1!)}$.
I'm unable to understand why are we only allowed to fill up each energy sub-level with a single particle. According to what I know, the Pauli Exclusion principle states that no two fermions can occupy the same quantum state, not energy level. For example, two electrons in the ground state of helium have the same energy, but due to opposite spins, they are in a different quantum state. Why are we not considering the spin while dealing with Fermi-Dirac Statistics.
If we ignore spin, then we can only use Energy levels ($n$) and angular momentum ($l\space$ and $m_l$), to define a quantum state, and then the formula is true, that no two fermions can occupy the same energy level ( quantum state ).
However, if we consider spin, the quantum state is no longer just defined by Energy value. It is also defined by the eigenvalue of the spin operator i.e. $m_s$. In that case, the number of particles that can be found in a particular sub-level of energy is $2s+1$. We can now find more than one fermions at a particular energy sublevel. If we consider this into Fermi Dirac statistics, how would our formula be modified?
Is my reasoning correct, or is there something obvious that I'm missing ?