Coefficiens binomialis

Numerus Coefficiens binomialis seu binominalis sic definitur:

$\begin{align} \binom nk &= \frac n1 \cdot \frac{n-1}2 \cdots \frac{n-(k-1)}k\\ &= \frac{n\cdot(n-1) \cdots (n-k+1)}{k!}\\ &= \prod_{j=1}^k \frac{n + 1 - j}j, \end{align}$

${k!}$ est factorialis numeri integri ${k}$.

Coefficientes binomiales sunt partes theorematis binomialis:

$(x+y)^n = \binom n0 x^n + \binom n1 x^{n-1} y + \ldots + \binom n{n-1} x y^{n-1} + \binom nn y^n = \sum_{k=0}^n \binom n k x^{n-k} y^{k}.$

Coefficientes binomiales perutilia sunt in arte combinatoria.