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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power into a sum involving terms of the form , where the exponents and are nonnegative integers with , and the coefficient of each term is a specific positive integer depending on and . For example, : The coefficient in the term of is known as the binomial coefficient or (the two have the same value). These coefficients for varying and can be arranged to form Pascal's triangle. These numbers also arise in combinatorics, where gives the number of different combinations of elements that can be chosen from an -element set. ==History== Special cases of the binomial theorem were known from ancient times. The 4th century B.C. Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2. There is evidence that the binomial theorem for cubes was known by the 6th century in India.〔〔 Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting ''k'' objects out of ''n'' without replacement, were of interest to the ancient Hindus. The earliest known reference to this combinatorial problem is the ''Chandaḥśāstra'' by the Hindu lyricist Pingala (c. 200 B.C.), which contains a method for its solution. The commentator Halayudha from the 10th century A.D. explains this method using what is now known as Pascal's triangle.〔 By the 6th century A.D., the Hindu mathematicians probably knew how to express this as a quotient , and a clear statement of this rule can be found in the 12th century text ''Lilavati'' by Bhaskara.〔 The binomial theorem as such can be found in the work of 11th-century Arabian mathematician Al-Karaji, who described the triangular pattern of the binomial coefficients. He also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using a primitive form of mathematical induction.〔 The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost.〔 The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui and also Chu Shih-Chieh.〔 Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, although those writings are now also lost.〔 In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express in terms of , via "Pascal's triangle". Blaise Pascal studied the eponymous triangle comprehensively in the treatise ''Traité du triangle arithmétique'' (1653). However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin.〔 Isaac Newton is generally credited with the generalised binomial theorem, valid for any rational exponent.〔 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 theorem」の詳細全文を読む スポンサード リンク
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