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Forming 2 teams of five from 11 people, but first person of each team has the captain role.

Mathematics Asked by diagoot on January 13, 2021

(a) A coach wants to form two (2) teams of five (5) from the eleven players on the team for a scrimmage game (i.e., just a small practice game where player positions are not important). The eleventh player will act as the referee. How many ways can the coach divide the team into two teams of five players?

(b) A coach wants to form two (2) teams of five (5) from the eleven players on the team for a scrimmage game, with the eleventh player again acting as the referee, but with a small change. The first person chosen for a team of five will be the captain of the team and will have extra responsibilities. For the rest of the players, their roles and positions are not important. How many ways can the coach divide the team into two teams of five players with one captain for each team?

For a), I have 11!/(5!5!2!) = 1386 ways. Dividing by 5! twice because there are two teams of five where the internal order doesn’t matter, and then 2! to ignore the order of the two teams. Finally, I ignored the last person (referee).

For b), I simply multiplied my answer in a) by 25: 1386 x 5 x 5 = 34650. Because there could be 5 permutations of the captain role in each of the two teams.

I’d appreciate it if someone can tell me if my reasoning/answers are correct, thanks.

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