Why Is Mixing Entropy of an Ideal Mixture Given by a Binomial Coefficient?

[superspartan9]

Homework Statement

Explain why, for an ideal mixture, the mixing entropy is given by
ΔS_mixing = k ln( Binomial Coefficient ( N, N_A )
where N is the total number of molecules and N_A is the number of molecules of type A. Use Stirling's Approximation to show that this expression is the same as the result from the previous problem when both N and N_A are large.

Homework Equations

ΔS = Nk ln(V_f/V_i)
Previous Solution: ΔS_mixing = -Nk[ x ln (x) + (1-x) ln (1-x) ]
Stirling Approx: N! ≈ N^Ne^-N√(2∏N)

The Attempt at a Solution

I originally thought it might mean mathematically derive, but it looks like that is more related to the second part of the problem. I have no idea how to get from the equations they give me to the binomial coefficient part of the solution. I believe I can do the second part if someone can provide some assistance to the explanation.

 

[BruceW]

for the first part, yes, I think you are just meant to explain. (not do anything mathematical). So, you've used the equation S = k ln(something) before, I'm guessing. What is that 'something' ? and why does it make sense that in this case, that 'something' is Binomial coefficient (N,N_A) ?