Cardinality of the set of finite multisets

Question

Consider a countably infinite set $S$.

The free commutative monoid on $S$ can be seen as the set $F(S)$ of all finite multisets of $S$.

Now is $F(S)$ also countably infinite, or is it larger? I think I'm missing some obvious idea.

Answer 1

It is countably infinite. For any $n\in\mathbb{N}$, the set $S^n$ is countable, and so $\bigsqcup_{n\in\mathbb{N}}S^n$ is countable (being a countable union of countable sets). But there is a surjection $\bigsqcup_{n\in\mathbb{N}}S^n\to F(S)$ (given a finite tuple of elements of $S$, you get a finite multiset by just taking the multiset of coordinates appearing in the tuple), so $F(S)$ is also countable.