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

${\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.$ For example, the fourth power of 1 + x is

{\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 ${\tbinom {4}{2}}={\tfrac {4!}{2!2!}}=6$ is the coefficient of the x2 term.

Arranging the numbers ${\tbinom {n}{0}},{\tbinom {n}{1}},\ldots ,{\tbinom {n}{n}}$ in successive rows for $n=0,1,2,\ldots$ gives a triangular array called Pascal's triangle, satisfying the recurrence relation

${\binom {n}{k}}={\binom {n-1}{k}}+{\binom {n-1}{k-1}}.$ The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. The symbol ${\tbinom {n}{k}}$ is usually read as "n choose k" because there are ${\tbinom {n}{k}}$ ways to choose an (unordered) subset of k elements from a fixed set of n elements. For example, there are ${\tbinom {4}{2}}=6$ ways to choose 2 elements from $\{1,2,3,4\},$ namely $\{1,2\},\,\{1,3\},\,\{1,4\},\,\{2,3\},\,\{2,4\},$ and $\{3,4\}.$ The binomial coefficients can be generalized to ${\tbinom {z}{k}}$ for any complex number z and integer k ≥ 0, and many of their properties continue to hold in this more general form.