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Race problem

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by zagcollins » Mon Jul 14, 2008 8:32 am
For a certain race, 3 teams were allowed to enter 3 members each. A team earned 6 – n points whenever one of its members finished in nth place, where 1 ≤ n ≤ 5. There were no ties, disqualifications, or withdrawals. If no team earned more than 6 points, what is the least possible score a team could have earned?

A. 0
B. 1
C. 2
D. 3
E. 4
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by Ian Stewart » Mon Jul 14, 2008 8:51 am
The top five finishers get 5, 4, 3, 2 and 1 points, respectively. That is, 15 points are awarded in total. To find the least points one team could get, we want the other teams to get the maximum possible, which is 6. If both the other teams get 6 points, the remaining team could get 3 points. We should do a quick check to make sure that the teams actually can get 6, 6, and 3 points: and indeed they can, if one gets 5+1, one gets 4+2, and the other gets 3.
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by zagcollins » Mon Jul 14, 2008 9:07 am
thanks, Ian...great explanation!
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Re: Race problem

by parallel_chase » Mon Jul 14, 2008 10:32 am
zagcollins wrote:For a certain race, 3 teams were allowed to enter 3 members each. A team earned 6 – n points whenever one of its members finished in nth place, where 1 ≤ n ≤ 5. There were no ties, disqualifications, or withdrawals. If no team earned more than 6 points, what is the least possible score a team could have earned?

A. 0
B. 1
C. 2
D. 3
E. 4
Answer D.

To find the minimum number of points a team could score, every teammate from that team should come last i.e. 5th since there is no disqualification or withdrawals.

6-5 + 6-5 + 6-5 = 3

Correct me if I am missing any part or strategy.

Thanks
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Re: Race problem

by Ian Stewart » Mon Jul 14, 2008 4:21 pm
parallel_chase wrote:
To find the minimum number of points a team could score, every teammate from that team should come last i.e. 5th since there is no disqualification or withdrawals.
No, the question says there are no ties. You can't have three people finishing 5th- only one person can finish 5th.
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by sudhir3127 » Mon Jul 14, 2008 7:20 pm
Isn't a tie from a team perspective rather than team mates..?
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by ksh » Mon Jul 14, 2008 11:57 pm
I can further simplify Ian's argument as:

given n is from1 through 5. assume the following table:
If a member finishes 1st,2nd,3rd... he scores
6-1=5 points
6-2=4
6-3=3
6-4=2
6-5=1

Also, there is no tie and no two players score same. assigning maximum 6 points to two teams i.e. (5+1) and (4+2) only 3 remains
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A simple way... is this correct though?

by dasp16 » Thu Mar 19, 2009 6:44 am
If there are no ties or anything the possible minimum score would be if 1 team come in last all the time :

team A
Place
(6-5) =1 point
(6-4) = 2 points
(6-3) = 3 points total 6 points

so the least possible place to get the six points required is 3

As I see it any combination will give a quantity higher than 6
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by sureshbala » Thu Mar 19, 2009 7:38 am
The top 5 performers of the race will get 5,4,3,2 and 1 points respectively. If we can give away all these 15 points to two teams (since there are only 5 members and 2 teams contain 6 members) so one team will get 0 points.

But it is given that a team can get a maximum of 6 points. So we can give only 6+6 =12 points to 2 teams and hence the remaining 3 points must be given to the left over team. In fact we can give 6 points each to 2 teams by giving 5+1 and 4+2 . Hence the left over 3 points were scored by the third team.
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Re: Race problem

by Caroline Lee » Sun May 03, 2009 12:32 am
zagcollins wrote:For a certain race, 3 teams were allowed to enter 3 members each. A team earned 6 – n points whenever one of its members finished in nth place, where 1 ≤ n ≤ 5. There were no ties, disqualifications, or withdrawals. If no team earned more than 6 points, what is the least possible score a team could have earned?

A. 0
B. 1
C. 2
D. 3
E. 4
It appears to me that the question should be corrected to: what is the least point(s) score a team could have earned?
Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind. ----Russell
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