Choice

There used to be a module for signing up for exams, which would do pretty much what you need. It allowed a group of students to choose an exam date from several available (projects in your case) with each date having a set limit for the number of students. It even had a waiting queue feature, if I recall. It ensured that a student selected only a single date. So, in your case, you could have set up 5 identical instances for students to select their projects. I don't think that module is supported anymore but I haven't looked at it for quite a while. It could be a good base to make your own custom module if needed.