# Definition:Binomial Coefficient/Real Numbers

## Definition

Let $r \in \R, k \in \Z$.

Then $\dbinom r k$ is defined as:

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

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

That is, when $k \ge 0$:

$\ds \dbinom r k = \dfrac {r \paren {r - 1} \cdots \paren {r - k + 1} } {k \paren {k - 1} \cdots 1} = \prod_{j \mathop = 1}^k \dfrac {r + 1 - j} j$

It can be seen that this agrees with the definition for integers when $r$ is an integer.

For most applications the integer form is sufficient.