# Help simplifying this using Stirlings' Formula?

1. Mar 6, 2015

Hey guys!

So I'm self studying Daniel Schroeder's Intro to Thermal Physics and we're to use Stirlings Approximation to simplify the following (assume that BOTH q and N are large, but we don't have a relationship between q and N):

$$\Omega \left( N,q\right) \approx \dfrac {\left( \dfrac {q+N} {q}\right) ^{q}\left( \dfrac {q+N} {N}\right) ^{N}} {\sqrt {2\pi q\left( q+N\right) / N}}$$

The formula to find the multiplicity of an Einstein solid is:

$$\Omega \left( N,q\right) =\dfrac {\left( N-1+q\right) !} {q!\left( N-1\right) !}$$

The hint in the text says to show that first:

$$\Omega \left( N,q\right) =\dfrac {N} {q+N}\dfrac {\left( q+N\right) !} {q!N!}$$

With q and N being large, I thought I could immediately say that

$$\Omega \left( N,q\right) \approx \dfrac {\left( q+N\right) !} {q!N!}$$

But I don't know where you get the

$$\dfrac {N} {q+N}$$

I tried immediately applying Stirlings Approximation but got nothing like the hint or the answer we're looking for.

I'm a little lost on where to begin here. Any thoughts?

Thanks so much.

2. Mar 6, 2015

### Mentallic

If you cancel the common factor of N in the numerator and N in the N! expression in the denominator, what do you get?

3. Mar 6, 2015