Definition 122 (Binomial coefficient, Dt.: Binomialkoeffizient) Let \( n \in \mathbb{N}_{0} \) and \( k \in \mathbb{Z} \). The binomial coefficient \( \left(\begin{array}{c}n \\ k\end{array}\right) \) of \( n \) and \( k \) is defined as follows:
$$ \left(\begin{array}{l} n \\ k \end{array}\right):=\left\{\begin{array}{cc} 0 & \text { if } \quad k<0, \\ \frac{n !}{k ! \cdot(n-k) !} & \text { if } \quad 0 \leq k \leq n, \\ 0 & \text { if } \quad k>n . \end{array}\right. $$
- Recall \( k !:=1 \) for \( k=0 \).
\( \mathbf{-} \) The binomial coefficient \( \left(\begin{array}{c}n \\ k\end{array}\right) \) is pronounced as " \( n \) choose k"; Dt.: "n über \( k \) "
Lemma 123 Let \( n \in \mathbb{N}_{0} \) and \( k \in \mathbb{Z} \).
$$ \left(\begin{array}{l} n \\ 0 \end{array}\right)=\left(\begin{array}{l} n \\ n \end{array}\right)=1 \quad\left(\begin{array}{l} n \\ 1 \end{array}\right)=\left(\begin{array}{c} n \\ n-1 \end{array}\right)=n \quad\left(\begin{array}{l} n \\ k \end{array}\right)=\left(\begin{array}{c} n \\ n-k \end{array}\right) $$
