Editing Binomial coefficient (section)

=== Generalization to infinite cardinals ===
The definition of the binomial coefficient can be generalized to [[Cardinal Number|infinite cardinals]] by defining:
: <math>{\alpha \choose \beta} = \left| \left\{ B \subseteq A : \left|B\right| = \beta \right\} \right|</math>
where {{mvar|A}} is some set with [[cardinality]] <math>\alpha</math>. One can show that the generalized binomial coefficient is well-defined, in the sense that no matter what set we choose to represent the [[cardinal number]] <math>\alpha</math>, <math display="inline">{\alpha \choose \beta}</math> will remain the same. For finite cardinals, this definition coincides with the standard definition of the binomial coefficient.

Assuming the [[Axiom of Choice]], one can show that <math display="inline">{\alpha \choose \alpha} = 2^{\alpha}</math> for any infinite cardinal <math>\alpha</math>.