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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク binomial coefficient ： ウィキペディア英語版 binomial coefficient In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by ''n'' and ''k'' is usually written $\backslash tbinom\; nk$. It is the coefficient of the ''x'' ''k'' term in the polynomial expansion of the binomial power (1 + ''x'') ''n''. Under suitable circumstances the value of the coefficient is given by the expression $\backslash tfrac$. Arranging binomial coefficients into rows for successive values of ''n'', and in which ''k'' ranges from 0 to ''n'', gives a triangular array called Pascal's triangle. This family of numbers also arises in many areas of mathematics other than algebra, especially in combinatorics. $\backslash tbinom\; nk$ is often read aloud as "''n'' choose ''k''", because there are $\backslash tbinom\; nk$ ways to choose ''k'' elements, disregarding their order, from a set of ''n'' elements. The properties of binomial coefficients have led to extending the meaning of the symbol $\backslash tbinom\; nk$ beyond the basic case where ''n'' and ''k'' are nonnegative integers with ; such expressions are still called binomial coefficients. ==History and notation== The notation $\backslash tbinom\; nk$ was introduced by Andreas von Ettingshausen in 1826, although the numbers were already known centuries before that (see Pascal's triangle). The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by Halayudha, on an ancient Sanskrit text, Pingala's Chandaḥśāstra. In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Lilavati.〔Lilavati Section 6, Chapter 4 (see ).〕 Alternative notations include ''C''(''n'', ''k''), ''n''''C''''k'', ''n''''C''''k'', ''C''''k''''n'', ''C''''n''''k'', ''C''''n'',''k'' in all of which the ''C'' stands for ''combinations'' or ''choices''. Many calculators use similar variants of the C notation as it can be represented on a single-line display. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「binomial coefficient」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.