Appendix A
Aspects of Combinatorial Analysis
1. Fundamental Principle of Counting: if some event can occur in n 1 different ways
and if, following this event, a second event can occur in n 2 different ways, and
so on, · · · , then the number of ways that the events can occur in that order is
n 1 n 2 · · · .
2. Permutations: Any arrangement of a set of n objects in a given order is called a
permutation. Any arrangement of any m ≤ n of the objects in a given order is
called an m-permutation, or a permutation of n objects taken m at a time. The
number of permutations of n objects taken m at a time is
P (n, m) =
n!
(n − m)!
.
(A.1)
3. Combinations: A combination of n objects taken m at a time irrespective of the
order is given by
C(n, m) =
P (n, m)
m!
=
n!
m!(n − m)!
.
(A.2)
A.1 Some Basic Probability Concepts
Consider 1 marble and 2 boxes, and ask ‘How can we distribute the marble between
the two boxes?’ Obviously, it can be done in two ways, i.e., in one box, or in the
other box.
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9
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