Seating with a gap

Three (distinct) people are to be seated in a row of 77 chairs so that no two of them occupy adjacent chairs. How many seatings are possible?

Show hints (2)+
  1. Lay out the 44 empty chairs first; they make 55 gaps to drop people into.
  2. Choose 33 of the 55 gaps, then order the distinct people: (53)3!\binom{5}{3}\cdot 3!.

Answer

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60

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Asked at: Timed Mental-Math & Sequences, Game-Based Aptitude

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