Coefficiens binomialis

Numerus Coefficiens binomialis seu binominalis sic definitur:

${\begin{aligned}{\binom {n}{k}}&={\frac {n}{1}}\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}}$${\begin{aligned}{\binom {n}{k}}&={\frac {n}{1}}\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!}$${k!}$ est factorialis numeri integri ${k}$${k}$.

Coefficientes binomiales sunt partes theorematis binomialis:

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

Coefficientes binomiales perutilia sunt in arte combinatoria.