Consider a box that is separated into two compartments by a thin wall. Each compartment has a volume V and temperature T. The first compartment contains N atoms of ideal monatomic gas A and the second compartment contains N atoms of ideal monatomic gas B. Assume that the electronic partition functions of both gases are equal to 1. The molecular partition function for each component is given by $$q_i = \frac{V}{Λ_i^3}$$
Firstly I am asked to write the total initial canonical partition function which is given in the question as $Q_{initial}=Q_AQ_B$. The algebra is fine, but why would the two separate gases have a total canonical function?
I was then asked to show that after the gases mix (and do not react) that the following is true: $$\frac{Q_{mixed}}{Q_{initial}}=4^N$$.
I got that $Q_{initial}=\frac{q_A^Nq_b^N}{(N!)^2}$. Therefore, the only reasonable expression for the mixed function is: $$Q_{mixed}=\frac{(2q_A)^N(2q_b)^N}{(N!)^2}$$
However, I cannot see why the individual molecular partition functions will have doubled. Especially since A and B are distinct gases.