1.26 Formulas for the binomial coefficients

The binomial coefficients can be written in terms of factorials

$${{n}\choose{1}}=n,$$

$${{n}\choose{2}}={{n(n-1)}\over{2!}},$$

$$\vdots\quad\quad\vdots$$

$${{n}\choose{k}}={{n(n-1)(n-2)\dots(n-k+1)}\over{1\cdot 2\cdot\dots\cdot k}}={{n!}\over{k!(n-k)!}}.$$

**

The formula

$${{\alpha}\choose{k}}={{\alpha(\alpha-1)(\alpha-2)\dots(\alpha-k+1)}\over{1\cdot 2\cdot\dots\cdot k}}.$$

**

$(**)$ for ${{\alpha}\choose{k}}$ makes sense for any real $\alpha$ and integer $k\geq 1$.