\problemdata                                                             
        {Scientists and secret keys} % latex-friendly label for the prob.
        {Counting} % The topic(s) of the problem content.  
        {Introduction to Combinatorial Mathematics, Page 8, Example 1-11} % Source (if known)                                          
        {S01 Tutorial 8, 4} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}           
Eleven scientists are working on a secret project.  They wish to lock up the documents in a cabinet such that the cabinet can be opened if and only if six or more of the scientists are present.

\begin{problemparts}
\problempart What is the smallest number of locks needed?

\solution{
We must have a group of six or more scientists present to open the safe.  Therefore, for any group of five scientists, there must be at least one lock they cannot open.  Moreover, for any two groups of five scientists, there must be two different locks they cannot open, because if both groups cannot open the same lock, there is a group of six scientists among these two groups who will not be able to open the cabinet.  Thus at least $\binom{11}{5}=462$ locks are needed.
}

\problempart What is the smallest number of keys to the locks each scientist must carry?

\solution{
Let A be one of the scientists.  Whenever A is associated with a group of five other scientists, A should have the key to the lock(s) that these five scientists were not able to open.  Thus, A carries at least $\binom{10}{5}=252$.  (Although we have only shown that these are lower bounds on the numbers of locks and keys, a scheme can actually be designed using these many locks and with each scientist carrying that many keys.)
}
                                 
\end{problemparts}
\end{problem}
