Picking Fair Swim Teams

A group of 9 kids want to hold a swimming competition. To start, each swimmer is given a rank from 1 to 9, with 1 being the fastest swimmer of the group and 9 being the slowest, without repetition.

Let swimmers 1, 2 and 3 be on Team \(A\), \(B\), and \(C\), respectively. These three swimmers each then pick two more kids for their team, making teams of 3.

The competition works as follows:

  • First, each team will face off against each other individually, meaning Team A will face Team B, Team B will face Team C, and Team C will face Team A. Afterwards, all three teams will compete against each other at the same time. Each of these \(4\) meets are scored separately and don't affect one another.

  • Each meet consists of 3 races. For each race, each team randomly selects a swimmer from their team who has yet to swim a race.

  • 1 point is awarded to the team with the winning swimmer from each race, and no points are given for 2nd and 3rd place.

  • Assume that a faster ranked swimmer will always beat a slower ranked swimmer.

At first it seems that the teams are picked fairly. In fact, the teams are picked such that Team A is more likely to beat Team B, Team B is more likely to beat Team C, and Team C is more likely to beat Team A! However, while a majority of the time the meet between all three teams will come out in a tie, one team has a higher probability to win than the other two, depending how each team picked their swimmers. If the probability of each team winning exactly 1 race each in the final meet is \(\frac{7}{12}\), then the team that is most likely to win has swimmers with the rankings \(R_1\), \(R_2\), and \(R_3\). What is \(R_1R_2R_3\)?


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