The Chemistry Maths Book, Second Edition

21.6 Permutations and combinations 609

The number of combinations of nobjects taken rat a time is the same as the number

of ways of dividing the nobjects into 2 groups of randn 1 − 1 r(2 boxes, one containing

robjects, the othern 1 − 1 r). More generally:

4.The number of ways of dividing ndifferent objects into kgroups, with n

1

objects

in group 1, n

2

in group 2, ..., and n

k

in group k, is the multinomial coefficient

(21.24)

This result is important in statistical thermodynamics when considering the number

of ways of distributing molecules amongst the available energy states. It gives rise to the

classical or Boltzmann statisticsthat describes the behaviour of many thermodynamic

systems.

Distinguishable and indistinguishable objects

The above theorems have been presented for sets of different or distinguishable

objects, and they must be modified or reinterpreted if some or all the objects are

indistinguishable. Two objects are called indistinguishable if their interchange

cannot be observed, and therefore does not give a new or distinct permutation or

combination. For example, theorem 4 can be reinterpreted for the number of distinct

permutations of nobjects made up of kgroups, the objects in any one group being

indistinguishable but different from the objects in all other groups.

EXAMPLE 21.10The distinct permutations of the 4 objects A, A, B, Bare

AABB, ABAB, ABBA, BAAB, BABA, BBAA

0 Exercises 19 –21

For n indistinguishable objects:

5.The number of permutations of nindistinguishable objects is 1.

6.The number of ways of distributing kindistinguishable objects (for example,

electrons) amongstn(≥ 1 k) boxes (quantum states) with not more than one object

per boxis.

The nboxes are either singly-occupied or are unoccupied, so that theorem 3 applies to

the boxes. This result is important in the statistical mechanics of systems of particles

that obey the Pauli exclusion principle. It gives rise to the quantum statisticscalled

Fermi–Dirac statisticsthat is important for the description of some thermodynamic

n

k

C

n

k

=

















2

4

6













=

n

nnn n

k

!

!!!!

123