Locks for a four-of-seven safe

A firm's 77 employees want a safe that opens only when at least 44 of them agree. They install locks, distributing keys so that every group of 44 can open all locks but no group of 33 can. What is the minimum number of locks required?

Show hints (2)+
  1. Each group of 33 must be locked out by some lock whose key they all lack.
  2. One lock per 33-subset (keys to the other 44): (73)=35\binom{7}{3}=35.

Answer

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35

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Asked at: Jane Street, Two Sigma

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