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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
$.


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  automatisch erstellt am 19.  8. 2013