In class today, it was announced that there would be 4 possible essay questions on an upcoming exam. Of those $4, 3$ would be chosen and we'd have to write on 2.
The teacher announced that, by his math, only $3$ would then have to be studied. The class responded that his math was incorrect and he seemingly soon realized his error. What is wrong? Why would studying only $3$ of the problems from the possible 4 be incorrect?
Looking at the combinations of $4$, {$1, 2, 3$}, {$1, 2, 4$}, {$1, 3, 4$}, and {$2, 3, 4$} are the only sets of three. By this I conclude that if I knew answers to only the first three of the essay questions, I would be guaranteed to answer at least two on the exam. What aren't I getting? Why would I still need to know all four essays for the exam? Thanks!
From 4 essays, the teacher will select 3, and of those you will have to write about 2. There are 4 possible ways for the teacher to give you the three essays:
So, if you study 1, 2, and 3 (or any of those 4 groups), then no matter which group is actually chosen, you will always find 2 essays on which to write, so your teacher is correct.