Using the same method as we did with Boltzmann Factors, we consider a system in thermal and diffusive equilibrium with another system.
P(N1,ε1)/P(N2,ε2) = g(N0 - N1, U0 - ε1)/g(N0 - N2, U0 - ε2)
And by the exact same method as our derivation of Boltzmann factors (no surprises here):
P(N1,ε1)/P(N2,ε2) = exp((N1μ - ε1)/T)/exp((N2μ - ε2)/T)
Which we will use in the exact same way to create the Grand Partition Function.