Theory Of Combinations, in mathematics, a statement of the laws which determine the possible variations in the grouping of any number of given signs. The signs and groups are known as elements and forms. There are three processes of combination. The first, which is termed permutation, consists in changing the order of the given elements so that the same arrangement is never repeated. The second, which is specially termed combination, consists in arranging the elements into partial groups, so that, without regarding the arrangement, precisely the same elements are not repeated in any form. In permutation, all the elements are contained in each form. In combination, each form may consist of two, three, or any other number of elements less than the whole number given. The third process, termed variation, is a union of the other two. It consists in first making all the forms possible by combination, and then multiplying each of these forms by permutation. In permutation there is a change in the order; in combination, in the contents or matter; and in variation, in both. The complication and possible number of forms is greatly increased when the elements are repeated.

The theory of combinations has application to ideas, sounds, colors, and even to food and other material compounds; but its principal use is in mathematical analysis and in the calculation of chances. The first important contribution to its development was by Buteo (1559), who represented all the throws possible with four dice. Pascal applied it to games, Mersenne to musical tones, and Guldin reckoned the number of words which could be formed from 23 letters. Leibnitz recognized its significance, and sought in vain to make use of it in discovering philosophical truths. Bernoulli and Euler labored upon it, but the first who gave it a scientific character was Hindenburg in 1778; and it was subsequently developed by Lagrange, Laplace, Poisson, Pfaff, Eschenbach, and Rothe. Among the treatises on the subject are the Lehrbuch der combinatorischen Analysis, by Weingartner (Leipsic, 1800-1801), and Vollstandiger Lehrbegriff der reinen Com-binationslehre, by Spehr (Brunswick, 1824).