Mathematics-Online course: Basic Mathematics-Combinatorics-Identities for Binomial Coefficients

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	Mathematics-Online course: Basic Mathematics - Combinatorics
	Identities for Binomial Coefficients

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We have the following identities for binomial coefficients:

$\bullet$ $\displaystyle 2^n$ $\displaystyle \sum_{k=0}^n \binom{n}{k}$ ,

$\bullet$ $\displaystyle 0$ $\displaystyle \sum_{k=0}^n \binom{n}{k} (-1)^k\,,\quad n \geq 1$ ,

$\bullet$ $\displaystyle \binom{n}{k}$ $\displaystyle \sum\limits_{i=0}^k \binom{n-k-1+i}{i}\,,\quad k < n$ ,

$\bullet$ $\displaystyle \binom{n}{k}$ $\displaystyle \sum\limits_{i=0}^{n-k} \binom{k-1+i}{i}\,,\quad k > 0$ .

(Authors: Höllig/Hörner/Walter)

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automatically generated 10/31/2008

$\bullet$	$\displaystyle 2^n$	$\displaystyle \sum_{k=0}^n \binom{n}{k}$ ,
$\bullet$	$\displaystyle 0$	$\displaystyle \sum_{k=0}^n \binom{n}{k} (-1)^k\,,\quad n \geq 1$ ,
$\bullet$	$\displaystyle \binom{n}{k}$	$\displaystyle \sum\limits_{i=0}^k \binom{n-k-1+i}{i}\,,\quad k < n$ ,
$\bullet$	$\displaystyle \binom{n}{k}$	$\displaystyle \sum\limits_{i=0}^{n-k} \binom{k-1+i}{i}\,,\quad k > 0$ .