Mathematik für Wirtschaftswissenschaftler by Prof. Dr. Volker Nollau (auth.)

By Prof. Dr. Volker Nollau (auth.)

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2 ..... 6 = 720 moglichen Reihenfolgen er wahlt (vgl. Fig. 19). I F • F: : . -' je 1 v Miiglichkeiten Fig. 19 Kombinationen Eine Auswahl von k verschiedenen Elementen ohne Beriicksichtigung der Anordnung aus der Menge {1, 2, ... , n} - also auch aus einer beliebigen n-elementigen Menge - heiftt eine Kombination (k-ter Ordnung). Die Anzahl C~k) (0 ~ k ~ n) aller Kombinationen (k-ter Ordnung) ist (~), d. h. C~k) = (~). 4 -- 15 Komb·mat·IOnen (4 . 0 r dnung ) , • F ur d. h. 15 MogIichkeiten aus {I, 2, ...

Man (A) ablesen, wobei auf der rechten Seite von (A) natiirlich hOchstens die ausgetauschten Yi vorkommen. Dabei sind wieder zwei Fane moglich: = O. Fall 2b: In (A) ist f,Lo '" o. Fall 2a: In (A) ist f,Lo Das lineare Gleichungssystem (G) ist also genau dann losbar, wenn der Fall 2b nicht eintritt.

N = 1 . 2·1·2· = ... ·k·l·2· ... ·(n-k) (k = 1,2, .. 3 Zahlenbereiche 35 (k= 1,2, ... ,n). Die beiden letztgenannten Eigenschaften fiihren zum Pascals chen Dreieck, bei dem (~) in der n-ten Zelle und k-ten Diagonale steht: k=O ,/ n=O n=1 n=2 n=3 n=4 1 1 1 1 1 1 4 k=2 ,/ 2 3 k=1 ,/ 1 1 3 k=4 ,/ 1 4 6 k=3 ,/ ....................................... • Aufierdem gilt fiir beliebige a, b E R und n ENder Binomische Lehrsatz: (a+b)n (~)anbo+ (~)an-lbl+ ... + = -. t (:)aOb n (n)an-kb k . ) Beispiele: • (a + b)4 = (~)a4bO + (~)a3bl + ...

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