Factors of Binomial Coefficient/Corollary 1

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Theorem

For all $r \in \R, k \in \Z$:

$\left ({r - k}\right) \dbinom r k = r \dbinom {r - 1} k$

from which:

$\dbinom r k = \dfrac r {r - k} \dbinom {r - 1} k$ (if $r \ne k$)


Proof

\(\displaystyle r \binom {r - 1} k\) \(=\) \(\displaystyle r \frac {\left({r - 1}\right) \left({\left({r - 1}\right) - 1}\right) \cdots \left({\left({r - 1}\right) - k + 2}\right) \left({\left({r - 1}\right) - k + 1}\right)} {k \left({k - 1}\right) \left({k - 2}\right) \cdots 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {r \left({r - 1}\right) \left({r - 2}\right) \cdots \left({r - k + 1}\right) \left({r - k}\right)} {k \left({k - 1}\right) \left({k - 2}\right) \cdots 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({r - k}\right) \frac {r \left({r - 1}\right) \left({r - 2}\right) \cdots \left({r - k + 1}\right)} {k \left({k - 1}\right) \left({k - 2}\right) \cdots 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({r - k}\right) \binom r k\)

$\Box$


Then:

$\dbinom r k = \dfrac r {r - k} \dbinom {r - 1} k$

follows from the

$\left ({r - k}\right) \dbinom r k = r \dbinom {r - 1} k$

by dividing both sides by $r - k$.

$\blacksquare$


Sources