It is shown in the article on Combinatorial Analysis that (w; 0,n) is the coefficient of a e z w in the ascending expansion of the fraction 1-a.

The general theory of polyhedra properly belongs to combinatorial analysis.

For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANTS; EQUATION; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROBABILITY.

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X3 (ll'1T2...) and it is seen intuitively that the number 0 remains unaltered when the first two of these partitions are interchanged (see Combinatorial Analysis).

The mathematical theory of probability and the allied theory of the combinatorial analysis were in effect created by the correspondence between Pascal and Fermat, concerning certain questions as to the division of stakes in games of chance, which had been propounded to the former by the gaming philosopher De Mere.

Algebraic Forms; Binomial; Combinatorial Analysis; Determin Ants; Equation; Continued Fraction; Function; Theory of groups; Logarithm; Number; Probability; Series.

See further COMBINATORIAL ANALYSIS.

Pfaff's researches bore chiefly on the theory of series, to which he applied the methods of the so-called combinatorial school of German mathematicians, and on the solution of differential equations.

The multi- (or poly-) nomial theorem has for its object the expansion of any power of a multinomial and was discussed in 1697 by Abraham Demoivre (see Combinatorial Analysis).