Binomialkoeffizient

Beispiel 1

1. \displaystyle \binom{n}{k} = \binom{n}{n-k}
   \displaystyle \binom{n}{n-k} = \dfrac{n!}{(n-n+k)!(n-k)!} = \dfrac{n!}{(k)!(n-k)!} = \binom{n}{k}
2. $\backslash binom\{n\}\{0\}\; =\; 1$
   \displaystyle \binom{n}{0} = \dfrac{n!}{(n-0)!0!} = \dfrac{n!}{n! \cdot 1} = \dfrac{n!}{n!} = 1
3. $\backslash binom\{n\}\{1\}\; =\; n$
   \displaystyle \binom{n}{1} = \dfrac{n!}{(n-1)!1!} = \dfrac{n! n }{(n-1)!n} = \dfrac{n!n }{n!} = n
4. \displaystyle \binom{n}{n} = 1
   \displaystyle \binom{n}{n} = \dfrac{n!}{(n-n)!n!} = \dfrac{n!}{0!n!} = \dfrac{n!}{1 \cdot n!} = 1
5. \displaystyle \binom{n}{n-1} = n
   \displaystyle \binom{n}{n-1} = \dfrac{n!}{(n-n+1)!(n-1)!} = \dfrac{n!}{1!(n-1)!} = \dfrac{n!n}{(n-1)!n} = \dfrac{n! n}{n!} = n
6. \displaystyle \binom{n-1}{k} + \binom{n-1}{k-1} = \binom{n}{k}
   \displaystyle \binom{n-1}{k} + \binom{n-1}{k-1} = \dfrac{(n-1)!}{(n-1-k)!k!} + \dfrac{(n-1)!}{(n-1-(k-1))!(k-1)!}
   \displaystyle = \dfrac{(n-1)!}{(n-1-k)!k!} + \dfrac{(n-1)!k}{(n-k)!(k-1)! k} = \dfrac{(n-1)!(n-k)}{(n-1-k)!k! (n-k)} + \dfrac{(n-1)! k}{(n-k)!k!}
   \displaystyle = \dfrac{(n-1)!(n-k)}{(n-k)!k!} + \dfrac{(n-1)!k}{(n-k)!k!} = \dfrac{(n-1)!(n-k) + (n-1)! k}{(n-k)!k!} = \dfrac{(n-1)!(n-k+k)}{(n-k)!k!}
   \displaystyle = \dfrac{(n-1)!n}{(n-k)!k!} = \dfrac{n!}{(n-k)!k!} = \binom{n}{k}