A combinatorial question?

Question

Eleven scientists are working on a secret project. They wish to lock up the documents in a cabinet so that the cabinet can be opened if and only if six or more of the scientists are present. What is the smallest number of locks needed? What is the smallest number of keys to the locks each scientist must carry?

Answer 1

1. How many locks? For each 5 scientist there must be a lock they cannot open, the number of locs is, hence, 11!/(5!*6!) = 462.

2. How many keys should each scientist carry? One must have all the keys for all the locks each five scientist out of the remaining 10 cannot open, the number of keys is 10!/(5!/*5!) =
= 252