Coefficiens binomialis

Numerus Coefficiens binomialis seu binominalis sic definitur:

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

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

Coefficientes binomiales sunt partes theorematis binomialis:

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

Coefficientes binomiales perutilia sunt in arte combinatoria.