Multiplicity function for N noninteracting spins

The multiplicity function Ω(n,N) is the same as the combinatoric function C(N,n). That is:

$\Omega (n,N)={N \choose n}={{N!} \over {n!(N-n)!}}$

It is primarily used in thermodynamics. If the multiplicity function represents a partially specified system of $N$ particles where $n$ are defined to be in one (of two) states, then the entropy of the system is given by the natural logarithm of the multiplicity function:

$\sigma =\ln \Omega (n,N)\,$