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Word Problems - Probability Word problem
Expert: Scott A Wilson - 10/22/2009
Question
If you can help with this problem also, it would be greatly appreciated. I have been banging my head with this for more than a month now with no success. I have been trying using Ven diagram, but it seems there are not enough info here:( I hope you can help.
On a soccer team there are four positions: goalie, back, midfielder, and forward. Each player on the team often plays more than one position during the course of the season.
On one soccer team there are 16 members. All of the goalies also play forward, and one goalie plays back as well. No goalies plays midfielders . There are as many backs as midfielders. The total numbers of midfielders is two-thirds the total number of forwards. Half of the team plays midfielder. No one plays only back.
Three people play three positions. Three people play only one position. The number of midfielders who play forward but not back is equal to the number of midfielders who play back but not forward and is equal to the number of midfielders who don't play anything else.
How many people play goalies?
How many people play back and forward but nothing else?
Thanks for help in advance.
Answer
Let G=goalie, B=back, M=midfielder, and F=forward.
The soccer team has 16 members.
Now to go through and make a condensed version of what is given.
1: F has all G, 1 G plays B.
2: No G play M.
3: B = M.
4: M = 2F/3.
5: M = 16/2.
6: No one is B alone.
7: 3 people play 3 positions.
8: 3 people play 1 position.
9: (M play B) = (M who play F) = (M who don't do anything else).
Now using 5:, we know that M is 8.
Using 3:, we also know that B is 8.
Using 4:, since M is 8, F is 12.
So far we have M=8, B=8, and F=12.
2: since M=8, says that G<=16-8, so G<=8.
7: says that there are 16-3-3=10 people who play 2 positions.
9: says that 1 or 2 people play MB / MF / M solo
By 2:, none of the M play G
This means that the M are playing 1 or 2 positions.
9: seems to lead to problems, since the M must play B, F, or nothing.
It also state that M who play B don't play F and M who play F don't play B.
Doesn't this divide the M in 3 equal sections?
This can't be, since 8 is not divisible by 3.
What am I missing?
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