Modern Engineering Thermodynamics

18.8 Quantum Statistical Thermodynamics 747
e. If there are four men and six women in the group, then the number of different unordered ten-person groups that can
be formed is given by Eq. (18.37) as
P4,6 10 = 10! =
10 × 9 × 8 × 7 × 6!
= 210 groups
4! × 6! 4 × 3 × 2 × 1 × 6!
Exercises
19. The available group in Example 18.7 suddenly drops from ten to six students from which to form the five-person
officer groups. How many different officer groups could you form without using any student more than once?
Answer: P5 6 5 = 720 groups:
20. Suppose you only need four officers instead of five in Example 18.7. How many different officer groups could you form
if you allowed students to be in more than one group? Assume ten students are still available for the officer positions.
Answer: P4 10 4 = 10,000 groups:
21. How many different arrangements are there of three black, seven red, and four green marbles?
Answer: P3,7,4 14 = 120,120 arrangements:
WHAT ARE PERMUTATIONS AND COMBINATIONS?
A permutation 5 is an arrangement of a group of items in specific order. Consider the group of three items denoted by X, Y,
and Z. The number of ordered arrangements of these three items taking two at a time without allowing repetition (this is
called permutations without repetition) is
P 3 2 = 3!
ð3!2Þ! = 6
These arrangements are XY, YX, YZ, ZY, XZ, andZX. However, the number of ordered arrangements when repetition is
allowed (called permutations with repetition) is
P 3 2 = 32 = 9
and these arrangements are XY, YX, YZ, ZY, XZ, and ZX plus the repeats XX, YY, and ZZ.
A combination is an arrangement of a group of items where the order does not matter. If order is not important within the
group, then the number of arrangements of the three items X, Y, andZ taking two at a time but not allowing items to be
repeated (called combinations without repetition) is
C 3 2 = 3!
ð3!2Þ!2! = 3
and these arrangements are XY, YZ, and XZ (note that XY is the same as YX when the order within the group is not important).
But, if we allow repetition of the items within the groups, then the number of arrangements (called combinations
with repetition) becomes
C 3 2
=
ð3 + 2!1Þ!
ð3!2Þ!2!
and they are XY, YZ, and XZ plus the repeats XX, YY, and ZZ.
In summary, if the order does not matter, it is a combination. If the order does matter, it is a permutation.
= 6
5 The word permutation is from the Latin “per” (thoroughly) + “mutare” (to change).
18.8 QUANTUM STATISTICAL THERMODYNAMICS
Webeginbydefiningthemicrostate of a group of molecules as the state produced by specifying the instantaneous
energy state of each molecule of the group. We define the macrostate as the instantaneous average state of
the collection of molecules, and the thermodynamic equilibrium state as being the most probable macrostate. The
mathematical probability of macrostate A is defined as
P A =
WðAÞ
∑ WðiÞ
i
(18.38)