There are fifty students in a class Each earned either an A

There are fifty students in a class. Each earned either an A, B, C, D or F on the final exam. How many final exams must I grade to ensure that at least four students have received the same grade?

Solution

We need to find minimum number of final exams to be graded so that at least four students received same grade.

So, let us start with grading 5 exams. So each could have a different grade as there are 5 grades.

So we grade 5 more exams. Now in worst case there are two exams each for each grade.

So we grade 5 more exams. In worst case there are 3 exams each for each grade.

So now grading one more exam will ensure there are at least 4 exams with same grade ie 4 students with the same grade.

So a minimum of 16 final exams need to be graded.