For identical bosons, the number is choose . To see
that think of the bosons as being inside a series of single
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.
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 on the bosons and from to on the separators between the boxes. There are then ways to arrange that total of objects. (There are choices for which object to put first, times choices for which object to put second, etcetera.) However, the different ways to order the subset of boson numbers do not produce different pictures if you erase the numbers again, so divide by . The same way, the different ways to order the subset of box separator numbers do not make a difference, so divide by .
For example, if 2 and 4, you get 5!2!3! or 10 system states.