Thermal Physics - energy, microstates, and probabilities

[Ascendant78]

Homework Statement

Homework Equations

The Attempt at a Solution

The first part I'm not worried about, but the second part is worked out in the "relevant equations" section. Honestly, it looks like more magic than a Harry Potter movie going on there to me. I'm at a loss as to what mathematical method/s are being utilized to get to that answer?

 

[stevendaryl]

I'm not sure that I understand your concern. From the equations, you can prove exactly that:

$\Omega(E=(r-s) \Delta) = \Omega(E=r \Delta)[\dfrac{r^s}{(N-r)^s}] [\dfrac{ (1-\frac{1}{r}) (1-\frac{2}{r}) ... (1 - \frac{s-1}{r})}{ (1 + \frac{1}{N-r}) (1 + \frac{2}{N-r}) ... (1 + \frac{s}{N-r})}]$

Then the only issue is proving that if $s \ll r$ and $s \ll (N-r)$, then the last factor is approximately 1.