home email


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.

John Wallis

Todhunter first cites the English mathematician John Wallis (1616-1703) whose de Algebra Tractatus was published in 1693. This work is prefaced by a history of the subject. Of interest here is the  material quoted from William Buckley. Wallis says:

"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. It was originally published as Arithmetica memoratiua, siue breuis, et compendiaria arithmeticae tractatio, etc. Londini: Ex officina Thomae Marshi by Gulielmus Buclaeus in 1567.

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
  1. Of the variety of Elections, or Choice, in taking or leaving One or more, out of a certain Number of things proposed.
  2. Of Alternations, or the different change of Order, in any Number of things proposed.
  3. Of the Divisors and Aliquot parts, of a Number proposed.
  4. 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.

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 preserving the Latin hexameter and permitting her virtues to exceed the numbers of the stars. 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.  See Gerardus Joannes Vossius' Opera Tome III  (1697) where on page 68 he mentions that there are 40,320 permutations of the words, and 1022 orderings which preserve meter. Also see Jean Prestet's  Élemens des mathématiques (1675) pages 342-348 who gives 2196. 

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.

Blaise Pascal

Advancement in the study of combinations occurred with the treatise on The Arithmetic Triangle by Blaise Pascal. This work was made available in 1665, but had been printed in 1654.

Franz van Schooten

On pages 373-403 of the Exercitationes Mathematicorum Schooten 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.

Gottleib Leibniz

In 1666 was published the Dissertatio de Arte Combinatoria 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.

Lesser Figures

Klau, Christoph. "Digressio de arenae numero." Christophori Clauii ... in sphaeram Ioannis de Sacro Bosco commentarius. (1581) Romae: Ex officina Dominici Base. English translation by G. Anderson (1784): "On the possibility of numbering the sands." The Arenarius of Archimedes, translated...with notes and illustrations. To which is added the dissertation of Christopher Clavius on the same subject, from the Latin. London: J. Johnson.

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.

Jean Prestet (1648-1691) has given in 1689 Nouveaux éléments de mathématiques in Volume I and  Volume II. In Volume I, Book V treats of Combinations and Permutations.