Subtraction formula for binomial coefficients
Formula
The following binomial identity holds. $$ \binom{m}{x} \left( {\frac{ m }{ x }} \right)^{-1} = \binom{m-1}{x-1} $$
Explanation
This identity is used when deriving the mean of the hypergeometric distribution.
Derivation
$$ \begin{align*} \binom{m}{x} \left( {\frac{ m }{ x }} \right)^{-1} =& {\frac{ m! }{ x! \left( m - x \right)! }} {\frac{ x }{ m }} \\ =& {\frac{ m! }{ x! \left( m - x \right)! }} {\frac{ x }{ m }} \\ =& {\frac{ (m-1)! }{ (x-1)! \left( m - x \right)! }} \\ =& {\frac{ (m-1)! }{ (x-1)! \left( (m-1) - (x-1) \right)! }} \\ =& \binom{m-1}{x-1} \end{align*} $$
■