Opening Boxes

Discrete Mathematics Level 3

Alice and Bob are in a room with 100 boxes. The boxes are labeled 1, 2, ..., 100. Bob has 100 sheets of paper, also labeled 1, 2, ..., 100. He randomly puts one sheet of paper into each box.

Alice may start by opening any box of her choice.

After opening a box and finding paper \(k\):

  • If box \(k\) is unopened, Alice will open box \(k\) and repeat this process.
  • If box \(k\) is open, Alice will stop.

What is the probability that Alice will open all of the boxes? When your answer is of the form \(\frac{p}{q}\), where \(p\) and \(q\) are positive, coprime integers, input \(p+q\).


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