TITLE: What is the probability of this exact same Champions League draw?
QUESTION [23 upvotes]: As you can see here, there has been a strange coincidence with the UEFA Champions League draw. The real draw which took place today, ended up being exactly the same as the rehearsal draw which took place yesterday.
Considering the following rules, what is the probability of this?
Group stage winners: 

Group A - Paris St. Germain (FRA)
Group B - Schalke 04 (GER)
Group C - Malaga (ESP)
Group D - Borussia Dortmund (GER)
Group E - Juventus (ITA)
Group F - Bayern Munich (GER)
Group G - Barcelona (ESP)
Group H - Manchester United (ENG)

Group stage runner-ups: 

Group A - Porto (POR)
Group B - Arsenal (ENG)
Group C - AC Milan (ITA)
Group D - Real Madrid (ESP)
Group E - Shakhtar Donetsk (UKR)
Group F - Valencia (ESP)
Group G - Celtic (SCO)
Group H - Galatasaray (TUR)

Winners from the group stage were seeded but they could not be drawn against a team who they played in the group stage, or another team from their association.

REPLY [4 votes]: Thanks for the very interesting and very elegant method using rook polynomials. As for me, poor computer scientist, I simply generated (by code) all possible draws, and there are indeed 5463 such draws. However, to get the exact probability of each draw, one needs to compute the probability of each team ordering, taking into account the way teams are selected. Indeed, all draws do not have equal probability. For the one that occurred, the probability is roughly 1/5515.