Consider a system of particles with allowed energy levels 
. Let 
be the number of allowed states at energy 
, and let 
be the actual number of particles at energy . The
values and are fixed, and the values are random according
to the particular arrangement of electrons. Two important physical
parameters are the total number of particles, 
, and the
total energy of the system, 
.
We impose the following hypotheses:
Given a specified 
, then the
conditional probability that a state at energy level is occupied is 
since exactly out of states
are filled; this is a consequence of hypotheses 1 and 3. By hypothesis 2, the probability 
that
the electron distribution is specified by 
is given by the ratio of 
, the number of such arrangements, to 
, the
total number of possible arrangements. By the total probability theorem:
Thus, 
is obtained as a weighted average of the
occupancy distribution conditioned on the various allowed values of .
Hypothesis 4 is valid as long as the number of energy levels,
states and particles is sufficiently large. This hypothesis permits us to
approximate the weighted average formula (1.5) for the
(unconditional) occupancy distribution very well with the term corresponding
to the most likely arrangement: 
. Thus, we need only determine the most likely arrangement and
study its properties in order to accurately characterize the system in
general.
Our task now is to compute , and to find 
which maximizes it. The number of ways to place indistinguishable
particles in states (hypothesis 3), with no more than one
particle in a single state (hypothesis 1), is:
and therefore:
To simplify the algebra to follow, instead of maximizing , we maximize 
; since the logarithmic
function 
is strictly monotonic increasing, the
maxima of W and 
occur at the same point . Thus:
We now invoke Stirling's approximation formula: for large n, 
, where the approximation is valid in the sense that the
ratio of both sides approaches 1 as 
. Thus,
We apply this approximation:
We now maximize (1.10) subject to the constraints that 
and 
are fixed, and that 
. We impose 
and 
using the methods of Lagrange multipliers: find and 
such that the function 
:
is maximized. We first locate critical points where 
for all i are 
, 
. The last two simply return the constraints imposed by
and . To compute 
, first note that 
. Thus:
Note that involves only , and not other 
's; the significance of the introduction of the Lagrange multipliers is that
the parameters 
and 
that appear in (1.12) are the same for all indices i; specifically, it is important to emphasize
that they do not depend on the energy level .
Letting 
denote the value of at which , we obtain:
and thus:
for some constants which do not depend on .
As a final remark, we often write 
instead of , thus suppressing the discrete nature of the allowed energy
levels. This corresponds to either approximating the range of discrete
levels with a continuum, considering the small energy difference between the
allowed levels, or to extending the results of this discrete analysis to the
case where a continuum of energy levels in indeed allowed.
