Negated Upper Index of Binomial Coefficient/Corollary 2

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Corollary to Negated Upper Index of Binomial Coefficient

Let $n, m \in \Z$.

Then:

$\dbinom n m = \paren {-1}^{n - m} \dbinom {-\paren {m + 1} } {n - m}$

where $\dbinom n m$ is a binomial coefficient.


Proof

\(\displaystyle \dbinom r k\) \(=\) \(\displaystyle \paren {-1}^k \dbinom {k - r - 1} k\) Negated Upper Index of Binomial Coefficient
\(\displaystyle \leadsto \ \ \) \(\displaystyle \dbinom n {n - m}\) \(=\) \(\displaystyle \paren {-1}^{n - m} \dbinom {\paren {n - m} - n - 1} {n - m}\) setting $r = n$ and $k = n - m$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \dbinom n m\) \(=\) \(\displaystyle \paren {-1}^{n - m} \dbinom {-m - 1} {n - m}\) Symmetry Rule for Binomial Coefficients
\(\displaystyle \) \(=\) \(\displaystyle \paren {-1}^{n - m} \dbinom {- \paren {m + 1} } {n - m}\)

$\blacksquare$


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