KöMaL 2006 B3935

Some characters in a Hamlet play are paired, for example, both members of a pair can play the role of Gertrude and the role of Player Queen. Before every performance it is decided by a lot who will play Gertrude and who Player Queen. Other pairs are also chosen by a similar lot. Sarah has already seen Hamlet, but she wants to see it again with a different actor playing the other member of the pair Gertrude/Player Queen, Claudius/Player King and Ophelia/Fortinbras, although not necessarily in one performance. How many tickets should she still buy to see all three roles in the other casting with 90% probability?