This is a question about thermal physics. There's this partition function Z = sum over all states s of the system ( exp(-E_s/T)). And its just used to calculate the probability of any state by taking the Boltzman factor exp(-E_s/T) of that state and dividing over the partition function. Theres one question that asks to show that the partition function for a combined system, Z(1and2) = Z(1)*Z(2). I understand the way its proved, you just take a double sum and say that E(1and2) =E(1)+E(2), so you can separate the sums. But by using a double sum arent you possibly overcounting some states? For example if E_s1 + E_s2 = 1 + 3 and E_s1 + E_s2 = 3 + 1, also 2+2... Shouldnt this just count as one state of the system, call it E_s = 4. Or would it be better to just keep it this way and then whenever you want to count the probability of observing an E = 4 of the double system, you would have to add all the possible boltzman factors corresponding to E = 4. I think i just answered my question..., but im just wondering whats the right way to think about it, because theres another part in the book about ideal gases talking about how when you have a system with distinct particles you can overcount, but when you have a system with identical particles, you have to multiply the partition function by 1/N! They also say at the end that in our argument we have assumed that all N occupied orbitals (i guess they mean energy levels) are always different orbitals. How does this change anything?