yes, you can ask questions in both interviewer and interviewee led cases. Good questions include:
- Those to understand how the business model of the client works
- Clarification of all the elements that were not clear to you in the prompt
- Goal clarification and constraints of the client to achieve it
You should ask questions until these points are clarified (usually 2-3 questions are enough - may depend on the complexity of the case).
If you are asking these types of questions and get the feedback you are asking too many questions, it may be you are asking questions in the wrong way - basically in a way such that the interviewer doesn't understand why you are asking the question, or feels you are trying to have the case solved by him.
For example, a good way to clarify the business model is:
"You told me our client is a major industrial goods company. Before moving to structure the problem, I would like to understand which is the current business model of the client - how do they make money basically. This will help to understand better how to face this problem."
In this way, you make clear that:
- You are going to structure later on
- You acknowledge the information already provided
- You align with the interviewer on the fact that this information will help to organize a better structure
A bad way to clarify the business model is:
"Which are our revenue streams?"
With such question, it's not clear for the interviewer if you will structure your approach, why you are asking such question and which connection it has with the information he/she provided you.
Hope this helps,