Factors of Binomial Coefficient/Corollary 2
Jump to navigation
Jump to search
Theorem
For all $r \in \R, k \in \Z$:
- $\paren {r - k + 1} \dbinom r {k - 1} = k \dbinom r k$
Proof
\(\ds \paren {r - k + 1} \dbinom r {k - 1}\) | \(=\) | \(\ds \paren {r - k + 1} \frac {r \paren {r - 1} \paren {r - 2} \cdots \paren {r - \paren {k - 1} + 1} } {\paren {k - 1} \paren {k - 2} \cdots 1}\) | Definition of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {r \paren {r - 1} \paren {r - 2} \cdots \paren {r - k + 1} } {\paren {k - 1} \paren {k - 2} \cdots 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds k \times \frac {r \paren {r - 1} \paren {r - 2} \cdots \paren {r - k + 1} } {k \paren {k - 1} \paren {k - 2} \cdots 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds k \dbinom r k\) | Definition of Binomial Coefficient |
$\blacksquare$