In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula which using factorial notation can be compactly expressed as For example, the fourth power of 1 + x is and the binomial coefficient is the coefficient of the x2 term. Arranging the numbers in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. The symbol is usually read as "n choose k" because there are ways to choose an (unordered) subset of k elements from a fixed set of n elements. For example, there are ways to choose 2 elements from namely and The binomial coefficients can be generalized to for any complex number z and integer k ≥ 0, and many of their properties continue to hold in this more general form. Andreas von Ettingshausen introduced the notation in 1826, although the numbers were known centuries earlier (see Pascal's triangle). In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī. Alternative notations include C(n, k), nCk, nCk, Ckn, Cnk, and Cn,k in all of which the C stands for combinations or choices. Many calculators use variants of the C notation because they can represent it on a single-line display. In this form the binomial coefficients are easily compared to k-permutations of n, written as P(n, k), etc. For natural numbers (taken to include 0) n and k, the binomial coefficient can be defined as the coefficient of the monomial Xk in the expansion of (1 + X)n. The same coefficient also occurs (if k ≤ n) in the binomial formula (valid for any elements x, y of a commutative ring), which explains the name "binomial coefficient".

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Related concepts (44)
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients arising in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, India, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row at the top (the 0th row). The entries in each row are numbered from the left beginning with and are usually staggered relative to the numbers in the adjacent rows.
Generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem.
Factorial
In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: For example, The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah.
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