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Problem of the week

In a graduation ceremony, \(n\) graduating students form a circle and their diplomas are distributed uniformly at random. Students who have their own diploma leave, and each of the remaining students passes the diploma she has to the student on her right, and this is one round. Again, each student with her own diploma leave and each of the remaining students passes the diploma to the student on her right and repeat this until everyone leaves. What is the probability that this process takes exactly \(k \) rounds until everyone leaves.