We wish to lock a vault with different locks. The vault committee has \(11\) members, each of whom has keys to some of the locks, but not all of them.
What is the smallest possible number of locks that we need to lock the vault so that each group of \(6\) members can open it together with the keys they have, but no group of just \(5\) members can ever do it? Note that a lock can have multiple keys that open it and a person can have keys to more than one lock.