D.25 Number of boson states

For identical bosons, the number is $I+N-1$ choose $I$. To see that think of the $I$ bosons as being inside a series of $N$ single particle-state boxes. The idea is as illustrated in figure D.2; the circles are the bosons and the thin lines separate the boxes. In the picture as shown, each term in the group of states has one boson in the first single-particle function, three bosons in the second, three bosons in the third, etcetera.

Figure D.2: Bosons in single-particle-state boxes.
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Each picture of this type corresponds to exactly one system state. To figure out how many different pictures there are, imagine there are numbers written from 1 to $I$ on the bosons and from $I+1$ to $I+N-1$ on the separators between the boxes. There are then $(I+N-1)!$ ways to arrange that total of $I+N-1$ objects. (There are $I+N-1$ choices for which object to put first, times $I+N-2$ choices for which object to put second, etcetera.) However, the $I!$ different ways to order the subset of boson numbers do not produce different pictures if you erase the numbers again, so divide by $I!$. The same way, the different ways to order the subset of box separator numbers do not make a difference, so divide by $(N-1)!$.

For example, if $I$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2 and $N$ $\vphantom0\raisebox{1.5pt}{$=$}$ 4, you get 5!$\raisebox{.5pt}{$/$}$​2!3! or 10 system states.