$\textbf{Seat on a Flight}$

There are $100$ airline passengers waiting in line to board a $100$-seat plane. For convenience, let the $n^{th}$ passenger in line hold a ticket for the $n^{th}$ seat. For some reasons, the first passenger decides to pick a random seat instead of his assigned seat (it is still possible that he or she picks the $1^{st}$ seat). Everybody will sit on his or her assigned seat unless this seat is occupied. In the latter case, that passenger will pick a random seat for himself or herself. Find the probability that the last passenger will sit on his or her assigned seat.

Derive the permutation formula $P_n^n=n\times (n-1)\times\cdots\times 2\times 1$ using the recursion method.