High Temperature Limit of Entropy in a Two Level System

[Kara386]

Homework Statement

Sounds like a physics problem but I'm sure of the physics, stuck on the maths. At high T a two level system has $\frac{N}{2}$ particles in each level. If entropy is given by $S = k\ln(\Omega)$, where $\Omega$ is the number of ways of getting $\frac{N}{2}$ particles per level, show the high temperature limit is $Nk\ln(2)$.

Homework Equations

The Attempt at a Solution

To the best of my knowledge, $Omega = \frac{N!}{(\frac{N}{2})!(\frac{N}{2})!}$. Taking $ln$ of this and using the Stirling approximation:

$N\ln(N) - N - [\frac{N}{2}\ln(\frac{N}{2}) - \frac{N}{2}] - [\frac{N}{2}\ln(\frac{N}{2})-\frac{N}{2}]$
$= N\ln(N) - N\ln(\frac{N}{2})$
$= -Nln(2)$
I've gone wrong with a minus sign somewhere but I really can't see where! Thanks for any help!

 

[DrClaude]

How do you write $-N \ln(N/2)$ as a sum of two logarithms?

 

[Kara386]

How do you write $-N \ln(N/2)$ as a sum of two logarithms?

Oh yes, there's the missing minus sign. Thank you! :)