# Picking Fair Swim Teams

**Discrete Mathematics**Level 5

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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