We begin by considering a general N-particle system with Hamiltonian
For simplicity, we consider the case that all the particles are of the same type. Having established the equivalence of the ensembles in the thermodynamic limit, we are free to choose the ensemble that is the most convenient on in which to work. Thus, we choose to work in the canonical ensemble, for which the partition function is
The 3N integrations over momentum variables can be done straighforwardly, giving
where 
is the thermal wavelength and
the quantity 
is known as the configurational partition function
The quantity
represents the probability that particle 1 will be found in a volume element 
at the point 
, particle 2 will be found in a volume element 
at the point 
,..., particle N will be found in a volume element 
at the point 
. To obtain the probability associated with some number
n;SPMlt;N of the particles, irrespective of the locations of the remaining
n+1,...,N particles, we simply integrate this expression over the
particles with indices n+1,...,N:
The probability that any particle will be found in the volume element
at the point and any particle will be found in
the volume element at the point ,...,any particle
will be found in the volume element 
at the point 
is defined to be
which comes about since the first particle can be chosen in N ways, the second chosen in N-1 ways, etc.
Consider the specal case of n=1. Then, by the above formula,
Thus, if we integrate over all , we find that
Thus, 
actually counts the number of particles likely to be
found, on average, in the volume element at the point .
Thus, integrating over the available volume, one finds, not surprisingly,
all the particles in the system.