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The staff of a bank includes a director, an assistant director, and four department heads. The director wants to have several locks on the strongroom door and several keys, so that:
· He can open the door alone.
· The assistant director can open it only if he is accompanied by any one of the department heads.
· Any three department heads can open it.
How many locks are needed, and how should the keys be distributed?
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It is simpler to consider which keys are not given to whom. The director gets the keys to all be considered further. Let A, B, C, D, and E be the assistant director and the four department heads. A gets all the keys but one, and B, C, D, and E all get the key A doesnt get. Now only the remaining keys of B, C, D, and E need be considered. Each group of two department heads should lack one key. There are six such groups, so seven keys are needed.
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