"Given that information, can we deduce with a GENERAL and STRAIGHTFORWARD(i.e to be programmed) way what all the correct answers are? I.e to find the string of 1's and 0's with length N, that validates the above information that had been given."
No, not always. For example, it might be that the n students all answer the m questions in exactly the same way. Knowing that each had i questions correct and m- i questions wrong tells us nothing about which answer were correct and which wrong.