SOLUTION: Ten friends want to play a game. They must be divided into three teams with three people in each team and one field judge. In how many ways can they do it?

Question 1120042: Ten friends want to play a game. They must be divided into three teams with three people in each team and one field judge. In how many ways can they do it?
Found 2 solutions by solver91311, greenestamps:
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

10 choose 3 ways to choose the first team, then for each of those choices there are 7 choose 3 ways to choose the second team. Then the product of those two numbers times 4 choose 3 ways to choose the third team, times 1 way to choose the judge.

Do the arithmetic.

John

My calculator said it, I believe it, that settles it

Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

The answer by the other tutor is obtained by choosing the three members of one team, then the three number of the second team, and then the three members of the third team, leaving the last person to be the field judge.

It would seem the answer would be different if you chose the three teams and the field judge in a different order. But it turns out that is not the case.

(1) Choose the field judge last:
C(10,3)*C(7,3)*C(4,3)*C(1,1) = 120*35*4*1 = 16800

(2) Choose the field judge after two teams:
C(10,3)*C(7,3)*C(4,1)*C(3,3) = 120*35*4*1 = 16800

(3) Choose the field judge after just one team:
C(10,3)*C(7,1)*C(6,3)*C(3,3) = 120*7*20*1 = 16800

(4) Choose the field judge first:
C(10,1)*C(9,3)*C(6,3)*C(3,3) = 10*84*20*1 = 16800

Here is another way to get the answer without having to worry about the order in which the teams are formed.

Imagine each person getting assigned to either team A, team B, team C, or the field judge F. Then

AAABBBCCCF would mean team A consists of players 1, 2, and 3; team B of players 4, 5, and 6; team C of players 7, 8,and 9; with player 10 the field judge.

AABCCBAFCB would mean team A consists of players 1, 2, and 7; team B of players 3, 6, and 10, team C of players 4, 5, and 9, with player 8 the field judge.

Each distinct arrangements of the letters AAABBBCCCF represents one of the possible ways the players can be divided into three teams of 3 and a field judge. By a well known counting principle, the number of different ways to do that is

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