610 Chapter 21Probability and statistics
systems at very low temperatures and, for example, for the description of the properties
of the conduction electrons in solids.
0 Exercise 22
Large numbers
EXAMPLE 21.11The number of permutations of 52 different objects (a pack of
cards) is52! 1 ≈ 1 8.1 1 × 110
67.
The number of objects considered in some applications of combinatorial theory in the
sciences can be very large. For example, we are concerned in statistical thermodynamics
with the possible arrangements of the particles of a mole of substance amongst the
available energy states, so thatn 1 ≈ 110
23, andn!is about 10
1025. The computation and
manipulation of such numbers is simplified by means of Stirling’s approximation
3(21.25)
where the symbol ~ is read as ‘asymptotically equal to’ and means that the ratio of
the two sides of (21.25) approaches unity asn 1 → 1 ∞. In many cases it isln 1 n!that is
of interest rather thann!itself. Then
(21.26)
Use of this asymptotic formula gives errors of about 12 (12n)inln 1 n!when nis large
(see Example 20.14). When nis large enough, we can put
ln 1 n! 1 ≈ 1 n 1 ln 1 n 1 − 1 n (21.27)
Some values are given in Table 21.6.
Table 21.6 Stirling’s approximation
n ln 1 n! eq.(21.26) n 1 ln 1 n 1 − 1 n ln 1 n! 1 − 1 (n 1 ln 1 n 1 − 1 n)
10 15.104 15.096 13.026 2.078
52 156.361 156.359 153.465 2.896
10
10
2.2 1 × 110
11
2.2 1 × 110
11
12.7
10
23
5.2 1 × 110
24
5.2 1 × 110
24
27.4
lnnnnn!−+∼ ln ln n
1
2
2 π
nn
n
e
n!
∼ 2 π
3James Stirling (1692–1770), Scottish mathematician, was, like Cotes and de Moivre, a friend of Newton.
The formula named for him appeared in his Methodus differentialisin 1730 and, also in 1730, in de Moivre’s
Miscellanea analytica.