Before we go on, I need to move away from our examples of binomial distributions and cast the general problem in terms more appropriate to Mechanics. We can always go back and generalize the paradigm15.9 but I will develop it along traditional lines.
The owner of the parking lot described earlier is only interested in the total number of cars parked because that number will determine his or her profit. In Mechanics the "coin of the realm" is energy, which we have already said is always written U in thermal physics. The abstract problem in STATISTICAL MECHANICS involves a complex system with many possible states, each of which has a certain total energy U. This energy may be in the form of the sum of the kinetic energies of all the atoms of a gas confined in a box of a certain volume, or it may be the sum of all the vibrational energies of a crystal; there is no end of variety in the physical examples. But we are always talking about the random, disordered energy of the system, the so-called internal energy, when we talk about U.
Now, given a certain amount of internal energy U,
the number of different fully-specified states of
the system whose total internal energy is U
[our partial constraint] is the conditional
MULTIPLICITY FUNCTION 
.
Taking the binomial distribution as our example again,
we could substitute crystal lattice sites
for "parking places" and defects for "cars"
[a defect could be an atom out of place, for instance].
If it takes an energy
to create one defect,
then the total internal energy stored in n defects
would be
.
Lots of other examples can be imagined, but this one
has the energy U proportional to the number n of
defects, so that you can see how the U-dependence
of 
in this case is just like the n-dependence
of
before.
So what?
Well, things start to get interesting when you put two such systems in contact so that U can flow freely between them through random statistical fluctuations.