Hard probability question/game-binomial dist.

Consider 50 people and series of 10 rooms. Each room has, for example, 5 doors that will open to the next room with known (totally known, including in later rooms), but different probabilities.

Everyone begins in room 1, and each person selects a door. If that door opens, all of the people that selected that door go into room 2. If not, they are all "out". the survivors then play the game in room 2 to get into room 3, and so on.

My goal is to allocate the people to the doors so to maximize the chance that at least one person gets through all of the rooms. The 2 people 2 room 2 door case is fairly easy to solve, and certainly there are several generalities like if p=1, everyone pick that door, but I'm looking for solutions or general principles for the harder cases.

Thanks very much for the help-this is not homework.