Factors of Binomial Coefficient/Corollary 2

Theorem

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

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

$\ds \paren {r - k + 1} \dbinom r {k - 1}$

$=$

$\ds \paren {r - k + 1} \frac {r \paren {r - 1} \paren {r - 2} \cdots \paren {r - \paren {k - 1} + 1} } {\paren {k - 1} \paren {k - 2} \cdots 1}$

$\ds $

$=$

$\ds \frac {r \paren {r - 1} \paren {r - 2} \cdots \paren {r - k + 1} } {\paren {k - 1} \paren {k - 2} \cdots 1}$

$\ds $

$=$

$\ds k \times \frac {r \paren {r - 1} \paren {r - 2} \cdots \paren {r - k + 1} } {k \paren {k - 1} \paren {k - 2} \cdots 1}$

$\ds $

$=$

$\ds k \dbinom r k$

$\blacksquare$