Assume that at the end of the semester there will be 30 students receiving grades for this class. Prove that some group of 3 students will get exactly the same letter grade (eg 3 students all earning an A-, or 3 students all earning an F). Prove that it is possible that no set of 4 students will get the same grade
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An awful example of a culture dependent question. – gnasher729 Sep 08 '14 at 22:05
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This is the pigeon-hole principle.
There are 12 possible grades: $A, A-, B+, B, B-, C+, C, C-, D+, D, D-, F$.
Assume by way of contradiction that no grade has 3 students associated with it. But this means that we were able to hand out only 24 grades. Since we have 30 students, we must give out at least one grade more that two times. So we must give out at least three of one grade in the class.
The following assignment of grades never gives out more than 3 grades of a given type: $$A: 3\\A-:3\\B+:3\\B:3\\B-:3\\C+:3\\C:3\\C-:3\\D:3\\F:3$$ We have given 30 total grades and none of them have been given 4 or more times.
Vladhagen
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