Consider whether some two students can have the same number of correct answers. (In other words, consider the case \(N = 2\).)

We can ignore the questions where the two students gave the same response. Let \(D\) be the number of questions where the two students gave different responses. Note that, in each of these \(D\) questions, exactly one student has the correct answer. When \(D\) is even, if one student has correct answers in \(D/2\) of these questions and the other has correct answers in the other \(D/2\) questions, they have the same number of correct answers. On the other hand, when \(D\) is odd, they never have the same number of correct answers.

Thus, two students never have the same number of correct answers if and only if their responses differ in an odd number of questions.

When the two students give different responses to a question, that question flips the parity of the difference of the numbers of times they choose \(1\). On the other hand, when the two students give the same response, this parity does not change.

Thus, the two students’ responses differ in an odd number of questions if and only if the numbers of times they choose \(1\) have different parities.

Therefore, two students never have the same number of correct answers if and only if the numbers of times they choose \(1\) have different parities.

The answer is the number of students who chose \(1\) odd numbers of times, multiplied by the number of students who chose \(1\) even numbers of times.