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 *x*^{k} term in the polynomial expansion of the binomialpower(1 + *x*)^{n}, and is given by the formula

For example, the fourth power of 1 + *x* is

and the binomial coefficient

${displaystyle {tbinom {4}{2}}={tfrac {4!}{2!2!}}=6}$ is the coefficient of the *x*^{2} term.

Arranging the numbers

${displaystyle {tbinom {n}{0}},{tbinom {n}{1}},ldots ,{tbinom {n}{n}}}$in successive rows for

${displaystyle n=0,1,2,ldots }$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

${displaystyle {tbinom {n}{k}}}$ 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

${displaystyle {1,2,3,4},}$namely

${displaystyle {1,2}{text{, }}{1,3}{text{, }}{1,4}{text{, }}{2,3}{text{, }}{2,4}{text{,}}}$and

${displaystyle {3,4}.}$The binomial coefficients can be generalized to

${displaystyle {tbinom {z}{k}}}$ for any complex number z and integer *k*≥ 0, and many of their properties continue to hold in this more general form.

## . . . Binomial coefficient . . .

Andreas von Ettingshausen introduced the notation

${displaystyle {tbinom {n}{k}}}$ in 1826,[1] although the numbers were known centuries earlier (see Pascal’s triangle). The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by Halayudha, on an ancient Sanskrit text, Pingala‘s *Chandaḥśāstra*. In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book *Līlāvatī*.[2]

Alternative notations include *C*(*n*, *k*), _{n}*C*_{k}, ^{n}*C*_{k}, *C*^{k}_{n}, *C*^{n}_{k}, and *C*_{n,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.

## . . . Binomial coefficient . . .

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