There is a risk here that the professor who only assigns 10 problems will only check one of them for correctness. If 5/10 of the answers are wrong but the professor verifies a correct answer for one of the 5/10 with a correct solution then their conclusion that the other 9/10 answers are correct due to some likelihood or probability function is invalid and a dimwit makes the grade.

Maybe professors should flip the script, instead of testing a student, have the student convince the professor they know the material. You will not pass until the professor feels confident.

Sounds a dissertation format maybe.

Presumably more questions can cover a wider variety of skills/techniques/topics. If the student doesn't know in advance which 10 will be selected, they either have to pray they're lucky, or work diligently on all problems.

presumably they glance at the other 90 too, and they then do a thorough verification on that random 10

the analogy is not great, but in cryptography something similar is at play (easy to get/check trivial properties and then hard to achieve/produce/fake the interesting ones)