Binomialkoeffizient I.39

$k\leq n,n\geq0,k\geq0$

$$\left(\begin{array}{c} n\\ k\end{array}\right) = \frac{n!}{k!(n-k)!}$$

$$= \frac{n\cdot\ldots\cdot(n-k+1)}{1\cdot\ldots\cdot(k-1)\cdot k}$$

$$\left(\begin{array}{c} n\\ k\end{array}\right)=\left(\begin{array}{c} n\\ n-k\end{array}\right)$$

$$\left(\begin{array}{c} n\\ k-1\end{array}\right)+\left(\begin{array}{c} n\\ k\end{array}\right)=\left(\begin{array}{c} n+1\\ k\end{array}\right)$$

$$\left(\begin{array}{c} n\\ 0\end{array}\right)=1\quad\left(\begin{array}{c} n\\ 1\end{array}\right)=n$$

• Binomischer Satz
  $(a+b)^{n}=\sum_{k=0}^{n}\left(\begin{array}{c} n\\ k\end{array}\right)a^{n-k}b^{k}$
  $$\sum_{k=0}^{n}\left(\begin{array}{c} n\\ k\end{array}\right)=2^{n}$$