TITLE: What is the minimum number of locks on the cabinet that would satisfy these conditions?
QUESTION [10 upvotes]: Eleven scientists want to have a cabinet built where they will keep some top secret work. They want multiple locks installed, with keys distributed in such a way that if any six scientists are present then they can open all the locks, but if only five are present then they cannot open all the locks. What is the minimum number of locks on the cabinet that would satisfy these conditions?

REPLY [8 votes]: I thought fiftyeight has a great and rigorous answer.
However, I think there is a shorter explanation/proof:
Given S #scientists and M size of subset - Since each subset of size M can't open all the locks, there should be just one lock (to impose minimality) that each such subset can't open. So we have a one-to-one / invertible mapping between subsets of size M and the locks. Number of subsets of size M is ${S \choose M}$. So the cardinality of set of locks is also ${S \choose M}$.