Multiplicity of a two state system

[d2x]

I only have a doubt about which definition to use for the multiplicity of a two state system. Clearly the total multiplicity of a two state system is given by:

$Ω=2^N$,

but what about the definition:

$Ω(N,n) = \binom{N}{n} = \frac{N!}{n!\cdot(N-n)!}$.

Clearly:
$2^N ≠ \frac{N!}{2!\cdot(N-2)!}$.

What is the difference between these two expressions for multiplicity? Is the second one incorrect for a two state system of N things?
Thanks.

 

[nasu]

If you sum the second one over n you get the first one.