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:
$\prod_i \frac{g_i^{N_i}}{N_i!}$

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:
$\prod_i \frac{(N_i+g_i-1)!}{N_i!(g_i-1)!}$

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.