See the article by Guetterman and Fetters, (2018) "Two Methodological Approaches to the Integration of Mixed Methods and Case Study Designs: A Systematic Review." American Behavioral Scientist, 62(7) 900-918.

Great question Emre Coban Emre Coban. Speaking from a research practitioner's experience, I think you need to look at the purpose of each approach and how you would use the methods. In a case study, you may use a single method or mixed methods to explore a specific phenomenon, a group, or event, to understand it and its context in depth.

In mixed methods, you would integrate qualitative and quantitative methods to achieve a comprehensive understanding of your broader research problem (research questions) that may relate to societal, cultural, environmental etc. issues.

Hi Emre Coban. Hopefully this will help.

A case study is a choice you make about the toy you want to study, and the people or places where it commonly is played with. For example...

- You aren't interested in other toys necessarily, just this one

- You want to see many dimensions of this single toy and its users (or maybe its producers, or customers)

- You may want to know about the context in which this toy is a toy, and not, say, a paper weight

As you prepare this case study, you may decide...

- You need to know who and how, and will want to get user experience like why it was fun (or not)

- You want to know how often the toy is played with, and maybe some demographic information like age, ethnicity, or accessibility status

Mixed methods research design (MMR) is like a pair of glasses you designate as the way to watch the people or places with that toy.

- Get a clearer view of how the toy is moved in space, how often it is moved, and by whom

- Draws your eye to the amount of people that can play with it at once, and read their diaries describing their experience

A case study designates what you will study, the limits (bounds) of the case, and the kinds of information you will try to utilize to prepare and interpret that study. It can be qualitative or quantitative. Perhaps most important to your question - it can also be both (multi-method) or integrated (mixed method). There are plenty of opinions about this depending on who you read about case studies (for example, Yin or Stake).

MMR is the way that you design for and get at that information. It may be appropriate for this toy, it might not. It will also guide how you combine and connect disparate portions of the methods you choose, and the data you derive. It cannot tell you what you want to study. It can be big or small. Like case studies, there is a lot of ink spilled about this area, and so who you prefer to learn from will influence its guiding principles and limitations (as exampled by David Morgan above).