Version numbers should be the ones you want. Here I do it with the last ones available at the moment of writing.
The simplest way to install elixir is using your package manager. Sadly, at the time of writing only Fedora shows
the intention to keep its packages up to date. There you can simply sudo dnf install erlang elixir and you are good to go.
Anyway, if you intend to work with several versions of erlang or elixir at the same time, or you are tied to
a specific version, you will need to compile it yourself. Then asdf is your best friend.