syntax - Why are these questions structured differently?

How many rooms does your apartment have?

How many rooms have carpeting?

I'm a native speaker of English teaching at a language school. Recently I was stumped by a question made by a student.
It was:

How come in the first sentence, the question is structured the "regular way" i.e. auxiliary (does) + present simple of the verb (have), whereas in the second sentence we don't use an auxiliary.

I have a feeling that it might be related to subject / object, but honestly I'm not sure. Any help would be appreciated.