A quiz has $8$ multiple choice questions with five answers for each (A, B, C, D, and E).
The teacher decided to give two versions of the quiz with the same questions and same responses but switched the order of the answers to create two different keys with no answers in common.
One of the students ended up getting $7$ of the $8$ questions wrong, but the $7$ she got incorrect matched the other answer key perfectly.
The teacher decided to use binomial probabilities to find the probability of randomly guessing this close or closer to the other answer key.
What did the teacher find out?