Well you probably need math to show that it's VERY improbable of getting them all wrong just by guessing.
Would you give somebody who got them all correct an A? Why would you do that? Because they knew the material well enough to answer all of the questions correctly. If they get them all wrong then it seems like they did in fact know the correct answer but they answered them all wrong on purpose. So they still knew the material well enough that they could have answered them all correct. Or maybe they thought they were answering them correctly but filled in the wrong circle on a scantron or something... It's silly but that does seem to be the implication.
I think you're confused on what X is? X is the observed number of successes. So inherently it must be an integer. I think what you're trying to describe is P(X = 0) which is related to but is different than X.
So that is exactly the thing I want to talk about. To evaluate such probabilities does not require normal approximation or even knowing Binomial distribution. All you need to know is the independent properties. I suggest you first to know what is random variable, Binomial distribution first. Then try to understand why people need to make use of normal approximation, and when you need such approximation.