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It is well-formulated. One could come up with instances of other NP-complete problems that have trivial solutions, like "What if there are only two cities and one road between them? Then the Traveling Salesman Problem isn't that hard!" [1]

If the dean gives you a decently sized list, figuring out such an accommodation is very difficult (it grows exponentially with the number of students).

[1] http://en.wikipedia.org/wiki/Travelling_salesman_problem

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To be precise, the best known solutions to the problem grow exponentially with the number of students. The P ?= NP question is (very roughly speaking) asking if better than exponential solutions exist.

Say he gives you a list of 300 students who can't go together in rooms. You say, good, I will take the remaining 100 students, and give them rooms. It even has unique solution :)

Any NP-Complete problem will have cases that are trivial. The complexity of the problems is how they grow in the worst case.

Correction: every natural NP-complete problem will have easy cases. You can construct problems for which there is no easy case, which is a shame since I thought at one point you could use this trait to seperate P and NP

Interesting. How do you need to define "easy" for this to work? Can you give an example of a problem without an easy solution? Has this class of problem been studied?

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