祝你好运!

In American Chinese restaurants, the standard "dessert" is a "fortune cookie" -- a thin, crisp shell inside which is placed a small slip of paper, usually containing an inspirational message of some sort (rather than any kind of prophecy). Sometimes on the reverse of the slip of paper there is a Chinese word or phrase, along with six "lucky numbers" from 1 to 60 (inclusive).

If you were to open three fortune cookies, the expected number of distinct lucky numbers you'd get can be written as \(\frac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a+b\).

Details and assumptions

  • Just as an example, the three fortunes pictured have 13 distinct lucky numbers.
  • The lucky numbers are distributed uniformly.
  • The six numbers on each fortune are distinct.
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