Seven seated in age order

Seven people of distinct ages sit down at random in the seven seats of a round table. What is the probability that they end up seated in increasing order of age around the table (in either the clockwise or counter-clockwise direction)? The probability is 1M\tfrac1M for an integer MM; find MM.

Show hints (2)+
  1. There are 7!7! seatings. How many are increasing clockwise? (Fix the youngest's seat; the rest follow.)
  2. Add the counter-clockwise-increasing seatings (disjoint), giving 1414 favorable.

Answer

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360

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Asked at: Jane Street, Citadel

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