1. $[\mathbf{3}]$ Five of James' friends are sitting around a circular table to play a game of Fish. James chooses a place between two of his friends to pull up a chair and sit. Then, the six friends divide themselves into two disjoint teams, with each team consisting of three consecutive players at the table. If the order in which the three members of a team sit does not matter, how many possible (unordered) pairs of teams are possible?
Answer: 5 Note that the team not containing James must consist of three consecutive players who are already seated. We have 5 choices for the player sitting furthest clockwise on the team of which James is not a part. The choice of this player uniquely determines the teams, so we have a total of 5 possible pairs.