Binomial coefficient

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 nk ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and is given by the formula

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

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.


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