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)
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.
Todhunter first cites the English mathematician John Wallis (1616-1703)
in 1693. This work is prefaced by a history of the subject. Of interest
here is the material quoted from William Buckley. Wallis
hac occasione, dum de Combinationibus agitur; hic subjungere Regulam
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
Annis 1671, 1672." (page 489)
work by Buckley is appended to the Dialectica
of John Seton published in 1584. The verses explaining arithmetic are
on pages 261-275.
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:
shall subjoin to this Chapter (as properly appertaining to this place,)
an Explication of the Rule of
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.
The Tot Tibi Variations
Apparently, some gentlemen of the 17th century studied permutations as
a diversion. By Erycius Puteanus (in Flemish, Vander Putten and in
French, Henry Dupuy (1574-1646)) we have the book Eryci
Puteani Peietatis Thaumata in Bernardi Bauhusii è Societate
Jesu Proteum Parthenium
Bernardus Balhusius of Belgium (Bernard Balhuis, 1575-1619), to whom
this refers, was a Jesuit priest who is most well-known for his 5 books
of epigrams. The most famous of these is the one composed in
honor of the Virgin Mary.
tibi sunt dotes,
Virgo, quot sidera caelo.
many qualities are yours, Virgin, as stars in the sky.
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
even higher to 3312.
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.
Advancement in the study of combinations occurred with the treatise on The
Arithmetic Triangle by Blaise
This work was made available in 1665, but had been printed in 1654.
Franz van Schooten
On pages 373-403 of the Exercitationes
introduces the section "Ratio inveniendi electiones omnes, quae fieri
possunt, data multitudine rerum." That is, the reckoning of discovering
all choices, which are able to happen, given from many things.
In 1666 was published the Dissertatio
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
Jean Prestet (1648-1691) has given in 1689 Nouveaux
éléments de mathématiques
I and Volume
II. In Volume I, Book V treats
of Combinations and 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,
375 - 383 concern permutations and combinations.