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There are N 3-dimensional quantum harmonic oscillators, so the energy for each one is:

[tex]E_i = \hbar \omega (\frac{1}{2} + n_x^i + n_y^i + n_z^i)[/tex]. What is the total number of states from energy E_0 to E, and what is the density of states for E?

## The Attempt at a Solution

In class they've shown that for a 1-D harmonic oscillator:

[tex] \Omega(E) = \frac{(\frac{E-E_0}{\hbar \omega} + N-1)!}{(\frac{E-E_0}{\hbar \omega})!(N-1)!} \delta E[/tex].

How did they get that? I Don't understand how it becomes (roughly) [tex] (\sum_i n_i + N) choose N [/tex]. And so I really have no idea how to generalize it to 3D (even though I guess it should be just the same).