200 Delegates and a Conference Hall

200 delegates attend a conference is held in a hall. The hall has 7 doors, marked \(A,B,\ldots, G\). At each door, an entry book is kept, and the delegates entering through that door sign it in the order in which they enter. If each delegate is free to enter any time and through any door he likes, how many different sets of seven lists could arise in all?

If the answer can be expressed an \(^nP_k\) for some positive integers \(n,k\), find (the minimium value of) \(n+k\).

Details and Assumptions:

  • Assume that every person signs only at his first entry.
  • \(\large ^nP_k=\dfrac{n!}{(n-k)!}\).

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