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A group of 11 scientists are working on a secret project, the materials of which are kept in a safe. They want to be able to open the safe only when a majority of the group is present. Subsequently the safe is provided with a number of different locks, and each scientist is given the keys to a certain number of these locks. How many of these locks are required, and how many keys must each scientist have?

I guess number of locks required should be 11C5 + 1. Can someone please verify this and also I can't figure out how to calculate number of keys

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marked as duplicate by Gerry Myerson, GNUSupporter 8964民主女神 地下教會, Namaste discrete-mathematics Mar 12 '17 at 13:26

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ There's an extra part in this question, regarding number of keys $\endgroup$ – uzumaki Sep 24 '16 at 12:07
  • $\begingroup$ Sorry, I have retracted my vote to close. You have seen the link now, so I will also remove my former comment. $\endgroup$ – drhab Sep 24 '16 at 12:10
  • $\begingroup$ @GerryMyerson From the comments above, it seems that the duplicate was already suggested, and OP has pointed out that he also wants the number of keys, which doesn't seem to be covered in the answers. $\endgroup$ – Arnaud D. Mar 12 '17 at 12:44
  • $\begingroup$ @Arn, I can't make head nor tail out of the comments, and there is certainly no clear reference there to any particular previous question. If OP has seen the other question, and can argue that the current question is not a duplicate, that should be where everyone can see it, not in some ambiguous or deleted comments. $\endgroup$ – Gerry Myerson Mar 12 '17 at 22:18
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Because there needs to be a majority of the $11$ scientists present, we need $6$ scientists to open all of the locks. now, take any $5$ scientists. for those $5$ scientists there needs to be some lock that they can't open. so there is at least one lock without a key for every combination of $5$ scientists. therefore we can just count the ways to choose $5$ scientists. so the answer is

$$\begin{pmatrix}11\\ 5\end{pmatrix}$$ locks.


now for when it comes to the number of keys each scientist needs, any particular scientist has to have the missing key for any combination of $5$ of the other $10$ scientists. scientist $A$ has to have keys to allow for all of the locks to be opened when he is pared with any of the other remaining scientists. so, we need $$\begin{pmatrix}10\\ 5 \end{pmatrix}$$ keys per scientist.

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