How many n-element subsets of real numbers are there

Question

I was wondering if anyone could show me how to express the cardinality of all n-element subsets of real numbers.

Answer 1

Let $[{\bf R}]^n$ be the set of $n$-element subsets of $\mathbf R$. Assumming $n > 0$, we have $|[{\bf R}]^n| \ge |{\bf R}| = 2^{\aleph_0}$. Conversely the map $f \colon [\mathbf R]^n \to \mathbf R^n$ defined by $f(\{x_1, ..., x_n\}) = (x_1, ..., x_n)$ where $x_1 < ... < x_n$ is injective. So $|[\mathbf R]^n| \le |\mathbf R^n| = |\mathbf R|^n = |\mathbf R| = 2^{\aleph_0}$. Thus $|[\mathbf R]^n| = 2^{\aleph_0}$.