Is it use open source Elasticsearch 7.10.2 in production environment?
Welcome to our community! 
It's not clear what you are asking sorry.
yes, you can use v7.10.2 in pro env, and you can use GET _license API to get your license.
The default is the basic version
There is an open source distribution available up to 7.10, but be aware this is a dead end that it is no longer supported or updated. Newer versions are available with a free basic license that can be used in production although there are some limitations.