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COMBINATION, in mathematics, is the variation or alteration of any number of figures, letters, colors, sounds, &c, in all the different manners possible. The parts combined are called elements,The doctrine of combination is that branch of mathematics which teaches the results arising from all possible combinations, and gives rules respecting them.Combinatory analysis is the application of the doctrine of combination to analysis, and constitutes a branch of science often very involved. A system of characters is appropriated to this purpose. Hindenburg, of Leipsic, in 1778, gave it the character of an independent science; and it has been of important service in relation to the higher branches of mathematics. (See Weingartner's Lehrbuch der combinatorischen Analysis, Leipsic, 1801, 2 vols.) Permutations are those combinations in which, each time, all the elements are used, and the object is to determine how often they change their place, for instance, abed, acbd,bdac, &c. The number of possible changes or combinations is found by multiplying the terms 1, 2, 3, &c. continually into each other; thus, 2X3 = 6; 6X4 == 24 ; 24X5 = 120, &c. Thus the combinations of five quantities amount to 120. The changes that may be rung on twelve bells amount to 479,001,600; and the twentyfour letters of the alphabet admit of 62,044,840,173,323,943,936,000 changes or combinations.