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Concept
Binomial theorem
Summary
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 polynomial (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example, for n = 4,
The coefficient a in the term of axbyc is known as the binomial coefficient or (the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where gives the number of different combinations of b elements that can be chosen from an n-element set. Therefore is often pronounced as "n choose b".
Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2. Greek mathematican Diophantus cubed various binomials, including . Indian mathematican Aryabhata's method for finding cube roots, from around 510 CE, suggests that he knew the binomial formula for exponent 3.
Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting k objects out of n without replacement, were of interest to ancient Indian mathematicians. The earliest known reference to this combinatorial problem is the Chandaḥśāstra by the Indian lyricist Pingala (c. 200 BC), which contains a method for its solution. The commentator Halayudha from the 10th century AD explains this method. By the 6th century AD, the Indian 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 first formulation of the binomial theorem and the table of binomial coefficients, to our knowledge, can be found in a work by Al-Karaji, quoted by Al-Samaw'al in his "al-Bahir".
Official source
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