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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, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. Visualisation of binomial expansion up to the 4th power. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.
The ordinary binomial coefficient () counts the r-combinations chosen from an m-element set. If one takes those m elements to be the different character positions in a word of length m , then each r -combination corresponds to a word of length m using an alphabet of two letters, say {0,1}, with r copies of the letter 1 (indicating the positions ...
The solution to this particular problem is given by the binomial coefficient (+), which is the number of subsets of size k − 1 that can be formed from a set of size n + k − 1. If, for example, there are two balls and three bins, then the number of ways of placing the balls is ( 2 + 3 − 1 3 − 1 ) = ( 4 2 ) = 6 {\displaystyle {\tbinom {2 ...
Several theorems related to the triangle were known, including the binomial theorem. Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients. [1] Pascal's triangle was known in China during the 11th century through the work of the Chinese mathematician Jia Xian (1010–1070).
in which form it is clearly recognizable as an umbral variant of the binomial theorem (for more on umbral variants of the binomial theorem, see binomial type). The Chu–Vandermonde identity can also be seen to be a special case of Gauss's hypergeometric theorem, which states that
As with the (non-q) Chu–Vandermonde identity, there are several possible proofs of the q-Vandermonde identity.The following proof uses the q-binomial theorem.. One standard proof of the Chu–Vandermonde identity is to expand the product (+) (+) in two different ways.
Applying the above theorem to our functional equation yields (with () = +): [] = [] (+) Via the binomial theorem expansion, for even , the formula returns . This is expected as one can prove that the number of leaves of a binary tree are one more than the number of its internal nodes, so the total sum should always be an odd number.