30.3 PERMUTATIONS AND COMBINATIONS
may imagine then(distinguishable) objects set out on a table. Each combination
ofkobjects can then be made by pointing tokof thenobjectsinturn(with
repetitions allowed). Thesekequivalent selections distributed amongstndifferent
but re-choosable objects are strictly analogous to the placing ofkindistinguishable
‘balls’ inndifferent boxes with no restriction on the number of balls in each box.
A particular selection in the casek=7,n= 5 may be symbolised as
xxx||x|xx|x.This denotes three balls in the first box, none in the second, one in the third, two
in the fourth and one in the fifth. We therefore need only consider the number of
(distinguishable) ways in whichkcrosses andn−1 vertical lines can be arranged,
i.e. the number of permutations ofk+n−1 objects of whichkare identical
crosses andn−1 are identical lines. This is given by (30.33) as
(k+n−1)!
k!(n−1)!=n+k−^1 Ck. (30.36)We note that this expression also occurs in the binomial expansion for negative
integer powers. Ifnis a positive integer, it is straightforward to show that (see
chapter 1)
(a+b)−n=∑∞k=0(−1)kn+k−^1 Cka−n−kbk,whereaistakentobelargerthanbin magnitude.
A system contains a numberNof (non-interacting) particles, each of which can be in
any of the quantum states of the system. The structure of the set of quantum states is such
that there existRenergy levels with corresponding energiesEiand degeneraciesgi(i.e. the
ith energy level containsgiquantum states). Find the numbers of distinct ways in which
the particles can be distributed among the quantum states of the system such that theith
energy level containsniparticles, fori=1, 2 ,...,R, in the cases where the particles are
(i)distinguishable with no restriction on the number in each state;
(ii)indistinguishable with no restriction on the number in each state;
(iii)indistinguishable with a maximumof one particle in each state;
(iv)distinguishable with a maximum of one particle in each state.It is easiest to solve this problem in two stages. Let us first consider distributing theN
particles among theRenergy levels,withoutregard for the individual degenerate quantum
states that comprise each level. If the particles aredistinguishablethen the number of
distinct arrangements withniparticles in theith level,i=1, 2 ,...,R, is given by (30.35) as
N!
n 1 !n 2 !···nR!.
If, however, the particles areindistinguishablethen clearly there exists only one distinct
arrangement havingniparticles in theith level,i=1, 2 ,...,R. If we suppose that there
existwiways in which theniparticles in theith energy level can be distributed among
thegidegenerate states, then it follows that the number of distinct ways in which theN