Prof. X decides to get his students to do his final grading, so he devises this plan. Initially, each student has their own exams. Prof. X picks a random pair of students and they swap exams, grade them, and swap back. Prof. X then picks another random pair and continues this process until every exam has been graded at least once. Some may be graded more than once, but Prof. X will never pick the same pair of students twice. Show that there must be two students who end up grading the same number of exams (assuming there are at least two students in the class).

For example, If the students are Alice, Bob, and Carol maybe Prof. X first picks Alice and Bob to swap and grade exams. Then perhaps he picks Bob and Carol. Then all three exams have been graded (Alice's and Carol's once each, and Bob's twice) and Alice and Carol both graded the same number of exams (one).