The computations of probabilities
often depend upon the proper
enumeration of outcomes. As such it requires the theory of
permutations and combinations as the foundation. This section outlines
the development of the theory. Of course, given n distinct objects, the number of
permutations is given by n!.
The number of ways to select k
objects from among these n,
that is, the number of combinations of size k, is C(n,k)
=n!/[k!(n-k)!].
Two important variations to these problems are to permit the
permutations to have a shorter length than n and to relax the requirement that
the objects appear to be distinct. Less obvious is the connection of
these counting problems to the number of divisors of a number.

"Libet,
hac occasione, dum de Combinationibus agitur; hic subjungere Regulam
Combinationis quam
habet Guilielmus
Buclaeus, Anglus, in
Arithmetica sua, versibus scripta, ante annos plus minus 190; quae ad
calcem Logicae Joahnnis Seatoni subjicitur,
in Editione Cantabrigiensi,
ante annos quasi 60. (sed medose:) Consonam Doctrinae de
Combinationibus supra traditae, quam ego publicis Praelectionibus
exposui Oxoniae,
Annis 1671, 1672." (page 489)

This
work by Buckley is appended to the Dialectica
of John Seton published in 1584. The verses explaining arithmetic are
on pages 261-275.

In The Doctrine of Combinations and Permutations published by Francis Maseres in 1795 may be found an English translation of the part of the Algebra entitled "Of Combinations, Alternations and Aliquot Parts" (pages 271-351). We quote the corresponding paragraph to the passage given above:

In The Doctrine of Combinations and Permutations published by Francis Maseres in 1795 may be found an English translation of the part of the Algebra entitled "Of Combinations, Alternations and Aliquot Parts" (pages 271-351). We quote the corresponding paragraph to the passage given above:

"I
shall subjoin to this Chapter (as properly appertaining to this place,)
an Explication of the Rule of
Combination,
which I find in Buckley's
Arithmetick, at the end of Seaton's
Logick, (in the Cambridge edition;) which (because obscure,) Mr. George
Fairfax (a Teacher of the Mathematicks then in Oxford,) desired me to
explain; to whom (Sept. 12, 1674,) I gave the explication under
written; Consonant to the doctrine of this Treatise, (which had been
long before written, and was the subject of divers public Lectures in
Oxford, in the years 1671, 1672.)"

The chapters of Wallis are

- Of the variety of Elections, or Choice, in taking or leaving One or more, out of a certain Number of things proposed.
- Of Alternations, or the different change of Order, in any Number of things proposed.
- Of the Divisors and Aliquot parts, of a Number proposed.
- Monsieur Fermat's Problems concerning Divisors and Aliquot Parts.

Tot
tibi sunt dotes,
Virgo, quot sidera caelo.

As
many qualities are yours, Virgin, as stars in the sky.

Puteanus
lists a total of 1022 permutations of the words. This number was chosen
to correspond to the number of stars in Ptolemey's catalog. While still
preserving meter, Wallis increased the number of permutations to 3096
and Jakob
Bernoulli even higher to 3312.

Another line

Rex,
Dux, Sol, Lex, Lux,
Fons, Spes, Pax, Mons, Petra, Christus

according to Balhuis admits 3,628,800 permutations while preserving
meter. Wallis corrected this to 3,265,920. In 1666 was published the Dissertatio
de ArteCombinatoria of Leibniz.
This is the earliest work of him connected to mathematics, but it is of
little interest. More properly it should be classified among his
philosophical works. We do note that Leibniz does include in Problem I:
"Dato numero et exponente complexiones invenire" the arithmetic
triangle and uses it to compute the number of combinations of various
sizes. In Problem VI: "Dato numero rerum, variationes ordinis invenire"
he computes the number of permutations of 24 objects and also
examines the permutation of phrases. For example, he quotes the lines
from Thomas Lansius of Tübingen (Thomas Lanß 1577-1657):

Lex, Rex, Grex, Res, Spes, Jus, Thus,
Sol, Sol (bona), Lux, Laus

Mars, Mors, Sors, Lis, Vis, Styx, Pus,
Nox, Fex (mala), Crux, Fraus

which contain 11 monosyllables and thus admit 11! = 39,916,800 permutations.

André Tacquet (1612-1660), a Flemish Jesuit, has given Arithmeticae Theoria et Praxis, published at Bruxellis in 1655. The edition linked is the corrected version dated 1683. Pages 49-51 treat briefly of permutations. In Book V, Chapter 8, pages 375 - 383 concern permutations and combinations.