Zurück zu Identitäten
( n k ) + ( n k − 1 ) = n ! k ! ⋅ ( n − k ) ! + n ! ( k − 1 ) ! ⋅ ( n − k + 1 ) ! {\displaystyle {n \choose k}+{n \choose k-1}={\frac {n!}{k!\cdot (n-k)!}}+{\frac {n!}{(k-1)!\cdot (n-k+1)!}}} = n ! ⋅ ( n + 1 − k ) k ! ⋅ ( n + 1 − k ) ! + n ! ⋅ k k ! ⋅ ( n − k + 1 ) ! = ( n + 1 ) ! k ! ⋅ ( n + 1 − k ) ! = ( n + 1 k ) {\displaystyle ={\frac {n!\cdot (n+1-k)}{k!\cdot (n+1-k)!}}+{\frac {n!\cdot k}{k!\cdot (n-k+1)!}}={\frac {(n+1)!}{k!\cdot (n+1-k)!}}={n+1 \choose k}}
