They're also susceptible to old-school cheating - sharing answers. When I was in college, multiple choice exams were almost extinct because students would form groups and collect/share answers over the years.
You can solve that but it's a combinatorial explosion.
A long time ago, when I handed out exams, for each question, I used to program my exam questions into a generator that produced both not-entirely-identical questions for each student (typically, only the numeric values changed) and the matching answers for whoever was in charge of assessing.
You can solve that but it's a combinatorial explosion.