# Binomial coefficients for infinite cardinalities [duplicate]

Let’s define $$C_\alpha^\beta$$ as the cardinality of the set of all subsets with cardinality $$\beta$$ of a set with cardinality $$\alpha$$:

$$C_\alpha^\beta = |\{T \subset S| |T| = \beta \}|$$

where $$|S| = \alpha$$.

It is easy to see, that $$C_\alpha^\beta$$ is well defined, and if $$\alpha$$ and $$\beta$$ are finite, then $$C_\alpha^\beta$$ becomes classical binomial coefficients.

It is also true, that $$C_\alpha^\beta = 0$$ for all $$\beta > \alpha$$.

Now one can see, that for infinite $$\alpha$$ and finite $$\beta$$ we have $$C_\alpha^\beta = \alpha$$.

It is also true, that for infinite $$\alpha$$ $$C_\alpha^\alpha = 2^{\alpha}$$:

$$|\{T \subset S| |T| = |S| \}| \geq |\{T \subset S| |T| < |S| \}|$$, as if $$|T| < |S|$$, then $$|S \setminus T| = |S|$$, for infinite $$S$$.

$$2^{|S|} = |\{T \subset S \}| = |\{T \subset S| |T| = |S| \}| + |\{T \subset S| |T| < |S| \}| \leq |\{T \subset S| |T| = |S| \}| \leq |\{T \subset S \}| = 2^{|S|}$$

However, I do not know the values of $$C_\alpha^\beta$$, where $$\alpha$$ and $$\beta$$ are both infinite and $$\beta < \alpha$$. Is there a way to find them too?