Formelsammlung Mathematik: Binomialkoeffizienten

${\displaystyle {\binom {n}{k}}}$

Für ${\displaystyle a\in \mathbb {C} }$ und ${\displaystyle k\in \mathbb {Z} }$:

${\displaystyle {\binom {a}{k}}:={\begin{cases}{\frac {a^{\underline {k}}}{k!}}&{\text{wenn}}\;k>0,\\1&{\text{wenn}}\;k=0,\\0&{\text{wenn}}\;k<0.\end{cases}}}$

Hierbei ist

${\displaystyle a^{\underline {k}}:=\prod _{j=0}^{k-1}(a-j).}$

Für ${\displaystyle a,b\in \mathbb {C} }$:

${\displaystyle {\binom {a}{b}}:=\lim _{x\to a}\lim _{y\to b}{\frac {\Gamma (x+1)}{\Gamma (y+1)\Gamma (x-y+1)}}.}$

Für ${\displaystyle n,k\in \mathbb {Z} }$ gilt:

${\displaystyle {\binom {n}{k}}+{\binom {n}{k+1}}={\binom {n+1}{k+1}},}$

bzw.

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

Sei ${\displaystyle n,k\in \mathbb {Z} }$ mit ${\displaystyle 0\leq k\leq n}$. Es gilt:

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

${\displaystyle {\binom {n}{k}}={\binom {n}{n-k}},}$

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