### D.25 Num­ber of bo­son states

For iden­ti­cal bosons, the num­ber is choose . To see that think of the bosons as be­ing in­side a se­ries of 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.

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.