Counting problem in statistical mechanics

Consider a system of $N$ molecules, and they can each possess either property A or property B. No molecule can have no property, or both properties at the same time. Let $x$ be the fraction of having property A, and $-nepsilon$ is the penalty the system incurs in energy when $n$ molecules exhibit property A, and the system incurs no penalty if a molecule exhibits property B.

I want to find the density of states $Omega$ for this system, and the relation between $x$ and the temperature $T$ of the system.

I know energy $E$ of the system is: $E = -Nxepsilon implies frac{dx}{dE} = -frac{1}{Nepsilon}$

$$Omega (E=-nepsilon) = frac{N!}{(N-n)!n!}$$
Because we are saying that out of the $N$ particles we have, $n$ of them have property A, and $N$ choose $n$ is the way we would pick $n$ particles to have a certain property.

We know $$S=klog Omega$$
And using Stirling’s approximation and some math,
$$S = -kNbig( (1-x)log (1-x) + xlog xbig)$$

Then I use Maxwell’s relationship to find $x$ as a function of $T$,
$$frac{partial S}{partial E} = frac{partial S}{partial x} frac{dx}{dE} = frac{1}{T}$$

Doing the inversion, we get $$x(T) = frac{e^{epsilon/kT}}{1+e^{epsilon/kT}}$$

The problem I have is, if I say $epsilon = alpha k$ where $alpha >0$, $x(T)$ cannot be a value less than $0.5$, as at $x=0.5$, $Trightarrow infty$. Does this make sense? Where am I going wrong?