Sharings into nonempty sets

We learn here how to count the number of distributions of different objects into distinguishable boxes in such a way that no box remains empty. For instance, assume a hostel has got 3 big rooms A, B, C. How many ways are there to distribute 14 students in the 3 rooms in such a way that each room has at least a student? Translated into mathematical terms this is counting the number of sharings of a finite set into a prescribed number nonempty sets.

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