# 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 ${\displaystyle {\tbinom {n}{k}}.}$ 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

${\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.}$

For example, the fourth power of 1 + x is

{\displaystyle {\begin{aligned}(1+x)^{4}&={\tbinom {4}{0}}x^{0}+{\tbinom {4}{1}}x^{1}+{\tbinom {4}{2}}x^{2}+{\tbinom {4}{3}}x^{3}+{\tbinom {4}{4}}x^{4}\\&=1+4x+6x^{2}+4x^{3}+x^{4},\end{aligned}}}

and the binomial coefficient ${\displaystyle {\tbinom {4}{2}}={\tfrac {4!}{2!2!}}=6}$ is the coefficient of the x2 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

${\displaystyle {\binom {n}{k}}={\binom {n-1}{k}}+{\binom {n-1}{k-1}}.}$

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 ${\displaystyle {\tbinom {n}{k}}}$ ways to choose an (unordered) subset of k elements from a fixed set of n elements. For example, there are ${\displaystyle {\tbinom {4}{2}}=6}$ ways to choose 2 elements from ${\displaystyle \{1,2,3,4\},}$ namely ${\displaystyle \{1,2\},\,\{1,3\},\,\{1,4\},\,\{2,3\},\,\{2,4\},}$ 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.