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Now suppose there are two types of particles differing in a parameter which appears in their Hamiltonians. So for a system consisting of ## N_1 ## particles of the first kind and ## N_2 ## particles of the second kind, the partition function is ## Q^{(1)}_{N_1}(V,T)Q^{(2)}_{N_2}(V,T)=\frac{[Q^{(1)}_1(V,T)]^{N_1}[Q^{(2)}_1(V,T)]^{N_2}}{N_1!N_2! h^{3(N_1+N_2)}} ##. Now if I take the limit that the parameter is the same for the two kinds of particles, I expect to get ##Q_{N_1+N_2}(V,T)##, but instead, I get ##Q^{(1)}_{N_1}(V,T)Q^{(2\to 1)}_{N_2}(V,T)=\frac{[Q^{(1)}_1(V,T)]^{N_1}[Q^{(2\to 1)}_1(V,T)]^{N_2}}{N_1!N_2! h^{3(N_1+N_2)}} ## which is different from what I expect because instead of ## (N_1+N_2)! ## in the denominator, it has ## N_1!N_2! ##. It seems that the factor introduced to solve the Gibbs paradox, is itself causing a paradox. What is wrong here? What is it that I don't see?

Thanks