One of my teachers gives his students essay finals. First, he tells us three numbers, A, B, and C. He gives us A essay questions to study before the test. He picks B essay questions to put on the test, and we must pick out C of them to answer. He tells us to study A-B+C of the given questions if we want to pass the final.

As a procrastinator, I only studied the night before. Luckily, some other students had taken the test a day early, and could tell me which of the questions the teacher had picked. Of course, the teacher would pick a different combination of questions to give to the rest of the students. After hearing which questions were given, I realized I needed to study N less questions than was necessary before!

Find N, generalizing to all possible A, B, and C.

(In reply to No solution by Jonathan Chang)

I am happy to clarify any ambiguosities you may find.

This is the real method used by one of my teachers. 

First, he picked a set (A) of essay questions to show to us.  You may not assume anything about A except that it is a positive integer.  He obviously cannot give us half an essay question, or less than zero.

During finals, he picks any subset (B) of the essay questions to put on the test.  We know how many questions will be on the test, but we don't know which ones.  Again, B is a positive integer, and cannot be greater than A.

When we see the final, we pick a subset (C) of the essay questions shown on the test.  We may pick however many we choose.  Again, C is a positive integer which cannot be greater than B.

You may assume that we only pass if we studied all the essay questions we answered.

The seniors take finals early, but the teacher gives them a different set of questions on the test (though he gives the same number).  For example, the seniors might have gotten questions 1, 2, and 3, while we get the questions 3, 4, and 5.



I think you should know that not every problem can be absolutely clear about what is wanted.  It is rarely worth the extra space to explain every detail and every assumption required.  What I mean is that you can't expect clarity from every problem.

Posted by Tristan on 2005-04-14 05:32:52