Floating Numbers in Combinations

Question

What could be the answer to

${\displaystyle {\binom {2.5}{2}}}$

is it defined or considered as $0$ or $1$?

Answer 1

The answer could use the gamma function:

$$\binom {2.5} 2=\dfrac{2.5!}{0.5!\times2!}=\dfrac{\Gamma(3.5)}{\Gamma(1.5)2}=\dfrac{2.5\times1.5}2=1.875.$$

Answer 2

Generalizing $n!=n\cdot(n-1)!$ to fractional numbers,

$$\frac{2.5!}{2!}=\frac{2.5\cdot1.5\cdot0.5!}{2\cdot0.5!}=\frac{15}8.$$

Answer 3

It's called Generalized Binomial Coefficient. For any $\alpha \in \mathbb{R}$ $$ \binom{\alpha}{2} = \frac{\alpha!}{(\alpha-2)!2!} = \alpha(\alpha-1) \frac{1}{2!} = (2.5)(1.5)\frac{1}{2!} = \frac{15}{8} $$ Note this also works for $\alpha <0$.

Answer 4

The generalised binomial coefficient $\binom{x}{k}$ ($x\in\mathbf R,\:k\in\mathbf N$) is $$\binom xk=\frac{x(x-1)\dots(x-k+1)}{k!}.$$