This is the binomial theorem for a positive integral index.
There are extensions of the binomial theorem, by means of which approximate calculations can be made of fractions, surds, and powers of fractions and of surds; the main difference being that the number of terms which can be taken into account is unlimited, so that, although we may approach nearer and nearer to the true value, we never attain it exactly.
With Descartes the use of exponents as now employed for denoting the powers of a quantity becomes systematic; and without some such step by which the homogeneity of successive powers is at once recognized, the binomial theorem could scarcely have been detected.
This property enables us to establish, by simple reasoning, certain relations between binomial coefficients.
These ideas are further developed in various papers in the Bulletin and in his L'Anthropometrie, ou mesure des differentes facultes de l'homme (18'ji), in which he lays great stress on the universal applicability of the binomial law, - according to which the number of cases in which, for instance, a certain height occurs among a large number of individuals is represented by an ordinate of a curve (the binomial) symmetrically situated with regard to the ordinate representing the mean result (average height).
The binomial theorem gives a formula for writing down the coefficient of any stated term in the expansion of any stated power of a given binomial.
Linnaeus' invention of binomial nomenclature for designating species served systematic biology admirably, but at the same time, by attaching preponderating importance to a particular grade in classification, crystallized the doctrine of fixity.
Application of Binomial Theorem to Rational Integral Functions.
Relation of Binomial Coefficients to Summation of Series.
A multinomial consisting of two or of three terms is a binomial or a trinomial.