# Binomial Coefficient/Examples/5 from -2

## Example of Binomial Coefficient

$\dbinom {-2} 5 = -6$

## Proof

From the definition of binomial coefficient:

$\dbinom r k = \begin {cases} \dfrac {r^{\underline k} } {k!} & : k \ge 0 \\ & \\ 0 & : k < 0 \end {cases}$

where $r^{\underline k}$ denotes the falling factorial.

Therefore:

 $\ds \dbinom {-2} 5$ $=$ $\ds \dfrac { {-2}^{ \underline 5} } {5!}$ $\ds$ $=$ $\ds \dfrac {-2 \times -3 \times -4 \times -5 \times -6} {5 \times 4 \times 3 \times 2 \times 1}$ Definition of Falling Factorial, Definition of Factorial $\ds$ $=$ $\ds -6$

$\blacksquare$