夫四、(35 points) A school has $n$ computers, each placed in one of $n$ rooms, with each room having a unique key. A training class has $m$ students $(m>n)$, and every evening exactly $n$ students enter the computer rooms for practical operation. To ensure that each person has one computer, what is the minimum number of keys that should be prepared and distributed to these $m$ students, so that no matter which $n$ people enter the computer rooms each evening, they can use the keys they have been given to open the door of one room and enter to practice, and according to the principle that those with fewer keys open the door first, each person can get exactly one room.