Factors of Binomial Coefficient/Corollary 1

From ProofWiki
Jump to navigation Jump to search

Theorem

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

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

from which:

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


Proof

\(\ds r \binom {r - 1} k\) \(=\) \(\ds r \frac {\paren {r - 1} \paren {\paren {r - 1} - 1} \cdots \paren {\paren {r - 1} - k + 2} \paren {\paren {r - 1} - k + 1} } {k \paren {k - 1} \paren {k - 2} \cdots 1}\)
\(\ds \) \(=\) \(\ds \frac {r \paren {r - 1} \paren {r - 2} \cdots \paren {r - k + 1} \paren {r - k} } {k \paren {k - 1} \paren {k - 2} \cdots 1}\)
\(\ds \) \(=\) \(\ds \paren {r - k} \frac {r \paren {r - 1} \paren {r - 2} \cdots \paren {r - k + 1} } {k \paren {k - 1} \paren {k - 2} \cdots 1}\)
\(\ds \) \(=\) \(\ds \paren {r - k} \binom r k\)

$\Box$


Then:

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

follows from the

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

by dividing both sides by $r - k$.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Factors_of_Binomial_Coefficient/Corollary_1&oldid=537290"