Question 184
The 7 dwarfs keep their treasure in the basement of the Škofjeloška castle. The treasure is behind 12 doors, and each door is locked with 12 locks. All locks are different. Each gnome has keys for several locks. Any 3 dwarves together have the keys to all the locks. Prove that the gnomes have a total of at least 333 (not necessarily different) keys.