why for k=8, N=15. N=11 looks perfectly correct.

If K=8,

i.e 1,2,3,4,5,6,7,8 students.

Now, each student has to sit for at least 1 lab of each professor.

For N=11,

Students 1 will sit in all labs, hence qualified for exam.

Students 2 will sit in labs: 2,4,6,8,10,12,14,16,18,20,22. hence qualified for exam. (At least 1 lab of all Prof covered)

Students 3 will sit in labs: 3,6,9,12,15,18,21,24,27,30,33 labs. hence qualified for exam. (At least 1 lab of all Prof covered)

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Similarly Student 8 will sit in 8,16,.....,88 labs, hence qualified for exam. (At least 1 lab of all Prof covered)

Please help, where I'm wrong.

Re: why for k=8, N=15. N=11 looks perfectly correct.

Just got it.

N=11 will violate "exactly K students" condition. For N=11, there will be 10 students who can qualify for exams.