I'm a computational chemist trying to understand more deeply the concepts of statistical mechanics. I followed Susskind's lectures. He starts by dividing a system in N subsystems; for my brain to like this approach and understand what comes next, I want to think about the subsystems as the actual particles (atoms/molecules) that make up the system. Each particle will be in state $i$, and if we ignore degeneracy the state will have a unique energy $E_i$. The occupation number $n_i$ identifies the number of particles in state $i$ and the probability of finding a particle in state $i$ is by definition $P_i = \frac{n_i}{N}$. Just one more piece of terminology (please correct me if I use anything incorrectly): I call microstate the collective state in which the particles are at any given time.
Then, each possible microstate corresponds to a particular set of occupation numbers; on the other hand, a set of occupation numbers might describe more than one microstate. This allows us to conclude that the most probable set of occupation numbers will be the one with the maximum number of ways of redistributing the particles among the energy states (while keeping the set ${n_{i}}$ fixed), that is the occupation numbers corresponding to the greatest number of equivalent microstates. The "number of rearrangements" is given by:
$$C = \frac{N!}{\prod\limits_{i} n_{i}!}$$
and by maximizing this quantity (this Wikipedia page goes through the same derivation) we arrive at a definition of the partition function as:
$$Z = \sum_{i} e^{-\beta E_{i}}$$
where, importantly, the $E_i$ are the energy states of the particles. The energy:
$$\sum_{i} P_{i} E_{i} = E$$
will be the average energy of a particle. In many other places, I've seen written that the $E_i$ are the values of the total energy of the system in a particular microstate. How are these things related? The total energy would be the sum of the individual particle energies (assuming ideal gas) and the total partition function would be the product of the individual partition functions . . . still, the derivation seems to make sense if I'm considering the energy states for each particle, rather than the total microstate energy.
Susskind goes on to state that $\beta$ is the inverse of the temperature, and he derives an equation for the Helmholtz free energy as a function of $Z$:
$$E - TS = -T\log Z$$ $$F = -T\log Z$$
. . . again, if we have defined $E$ to be the average energy of a particle, does this equation still make sense?
If you have taken the time to read all this and plan on taking more to answer, first of all thank you, but secondly please answer in a way that doesn't use complex mathematical constructions that a chemist might not be familiar with.