A group of five people want to put a set of locks on a chest and distribute keys to the locks amongst themselves in such a way that all the locks on the chest could be opened only when at least three of them were present to open it.
How many locks would be needed, and how many keys?
Assuming that (i) each lock is locked and unlocked by one key, of which there can be any number of copies, and that (ii) all locks need to be unlocked to open the box, the answer is: Use one lock for each 3-person subset of the group keys for each lock being given to all people in its subset. This uses C(5,3)=10 locks and 10*3=30 keys.
ans: each member 6 keys
It can be easily shown that this is a minimum.
Edited on March 3, 2004, 5:47 pm