Mathematik-Online-Kurs: Mathematische Grundlagen-Kombinatorik-Binomischer Lehrsatz

$\displaystyle (a+b)^n$	$\displaystyle =$	$\displaystyle a^n + \left( \begin{array}{c} n \\ 1 \end{array}\right) a^{n-1}b ... ...^2 + \cdots + \left( \begin{array}{c} n \\ n-1 \end{array}\right)ab^{n-1} + b^n$
	$\displaystyle =$	$\displaystyle \sum_{k=0}^n \left( \begin{array}{c} n \\ k \end{array}\right)a^{n-k}b^k .$

$\displaystyle (a+b)^2$	$\displaystyle =$	$\displaystyle a^2+2ab+b^2\,,$
$\displaystyle (a+b)^3$	$\displaystyle =$	$\displaystyle a^3+3a^2 b+3ab^2 + b^3\,.$

(Aus: Lineare Algebra und Geometrie, Kimmerle)

$\displaystyle \sum_{k=0}^0 \left( \begin{array}{c} 0 \\ k \end{array}\right)a^{... ...d \sum_{k=0}^1 \left( \begin{array}{c} 1 \\ k \end{array}\right)a^{1-k}b^k=a+b$

$\displaystyle (a+b)^{n+1}$	$\displaystyle =$	$\displaystyle (a+b) \, (a+b)^n = (a+b) \,\sum_{k=0}^n \left( \begin{array}{c} n \\ k \end{array}\right) a^{n-k}b^k$
	$\displaystyle =$	$\displaystyle \sum_{k=0}^n \left( \begin{array}{c} n \\ k \end{array}\right) a^... ...+ \sum_{k=0}^n \left( \begin{array}{c} n \\ k \end{array}\right) a^{n-k}b^{k+1}$
	$\displaystyle =$	$\displaystyle a^{n+1} + \sum_{k=1}^n \left( \begin{array}{c} n \\ k \end{array}... ...n-1} \left( \begin{array}{c} n \\ k \end{array}\right) a^{n-k}b^{k+1} + b^{n+1}$
	$\displaystyle =$	$\displaystyle a^{n+1} + \sum_{k=1}^n \left( \begin{array}{c} n \\ k \end{array}... ...1}^n \left( \begin{array}{c} n \\ k-1 \end{array}\right) a^{n-k+1}b^k + b^{n+1}$
	$\displaystyle =$	$\displaystyle a^{n+1} + \sum_{k=1}^n \left[ \left( \begin{array}{c} n \\ k \end... ...eft( \begin{array}{c} n \\ k-1 \end{array}\right)\right] a^{n+1-k}b^k + b^{n+1}$
	$\displaystyle =$	$\displaystyle \sum_{k=0}^{n+1} \left( \begin{array}{c} n+1 \\ k \end{array}\right) a^{n+1-k}b^k$

(Aus: Lineare Algebra und Geometrie, Kimmerle)

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	Mathematik-Online-Kurs: Mathematische Grundlagen - Kombinatorik
	Binomischer Lehrsatz