How Do We Correctly Calculate the Distribution of Bosons Among Energy Levels?

[ShayanJ]

I want to calculate the number of ways for putting N_1,N_2,N_3,...,N_i,... bosons in energy levels with degeneracies g_1,g_2,g_3,...,g_i,....
The particles are indistinguishable and there can be any number of particles in a state.
The level with degeneracy g_i has N_i particles in it.The first of this N_i particles has g_i states to choose.The second,again has g_i choices and the same for all of them.So there are g_i^{N_i} ways for putting N_i particles in g_i sates.But the particles are indistinguishable so their order is not important and so g_i^{N_i} reduces to \frac{g_i^{N_i}}{N_i!}. So the number of ways for putting N_1,N_2,N_3,...,N_i,... bosons in energy levels with degeneracies g_1,g_2,g_3,...,g_i,... is:
<br /> \prod_i \frac{g_i^{N_i}}{N_i!}<br />

But the above result will give us sth like Boltzmann distribution,not Bose-Einstein's and we know that the answer should be like below:
<br /> \prod_i \frac{(N_i+g_i-1)!}{N_i!(g_i-1)!}<br />

But what is wrong?
In deriving my formula,I assumed only that the particles are indistinguishable and don't follow Pauli's principle,the same assumptions made for bosons.So what was different?
Thanks

 

[DrDu]

Dividing by N_i! you are overcounting states. Take just one state with degeneracy g_1=1 and two bosons.
Obviously dividing by 2! is not correct.