Question 579315: If there are a total of 14 players to choose from, how many ways can you pick your players if each team picks one at a time? (order does not matter)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! assume only 4 players.
assume the players are a,b,c,d
there are 6 teams composed of 2 players each that can be generated from a pool of 4 players.
those teams are:
ab
ac
ad
bc
bd
cd
any one of these combinations can be chosen by either team leader depending on what the other team leader chose.
the team leader that picks first gets 4 * 2 = 8 possible choices.
the team leader that picks second gets 3 * 1 = 3 possible choices.
the team leader who chooses first has a greater choice, but each team winds up with 2 players each regardless and the number of unique possible teams that can be formed is the same regardless of who picks first.
the formula for generating a team of 2 players out of 4 players is the combination formula of C(4,2) which becomes (4!) / (2!*2!) which becomes (4*3*2!) / (2*1*2!) which is equal to (4*3) / (2*1) which is equal to 6.
the formula for generating a team of 7 players out of 14 players is the combination formula of C(14,7) which becomes (14!) / (7!*7!) which becomes (14*13*12*11*10*9*8*7!) / (7*6*5*4*3*2*1*7!)
the 7! in the numerator and the denominator cancel out and you are left with (14*13*12*11*10*9*8) / (7*6*5*4*3*2*1)
this winds up being equal to 3432.
3432 possible teams of seven players each can be generated from 14 players where each team contains 7 players.
the team leader who gets to choose first gets more choices than the team leader who gets to pick second.
this works out as follows:
team leader that picks first gets to choose:
14 * 12 * 10 * 8 * 6 * 4 * 2 = 645120 possible choices.
team leader that picks second gets to choose:
13 * 11 * 9 * 7 * 5 * 3 * 1 = 135135 possible choices.
there is a definite advantage to being able to choose first.
regardless of who picks first, the possible teams that can be generated that are unique where order doesn't count are calculated using the combination formula of C(14,7).
this is what i believe is the correct way to calculate this.
the problem statement is:
If there are a total of 14 players to choose from, how many ways can you pick your players if each team picks one at a time? (order does not matter)
the question is difficult to answer because it is ambiguous.
do they want the number of possible teams that can be generated that are unique or do they want the number of possible ways to choose the team?
the number of possible ways to pick a team are different depending on who chooses first and who chooses second.
the number of possible unique teams that can be formed is the same regardless of who picks first and who picks second.
i would go with the combination formula because that's what you are studying and they didn't differentiate between who picks first and who picks second in the question.
that's just a guess though.
i have no idea what your instructor really wants.
order does not matter means combination.
order does matter means permutation.
i think combination is what applies here.
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