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# Formelsammlung Mathematik: Binomialkoeffizienten

### Wertetabelle

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

k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
n=0 1
n=1 1 1
n=2 1 2 1
n=3 1 3 3 1
n=4 1 4 6 4 1
n=5 1 5 10 10 5 1
n=6 1 6 15 20 15 6 1
n=7 1 7 21 35 35 21 7 1
n=8 1 8 28 56 70 56 28 8 1

### Definition

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

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

### Rechenregeln

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:

${\displaystyle {\binom {n}{k}}={\frac {n!}{k!\cdot (n-k)!}},}$
${\displaystyle {\binom {n}{k}}={\binom {n}{n-k}},}$
${\displaystyle {\binom {n+1}{k}}={\frac {n+1}{n-k+1}}{\binom {n}{k}}.}$