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For a certain race, 3 teams were allowed to enter 3 members each.

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caiyun

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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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KarishmaB

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05 Dec 2010, 19:44

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I3igDmsu

I am struggling with this one, it may be that I don't understand the question and information given.

If there are 3 teams of 3 members each (9 people total), wouldn't the possible finishes for each member of the teams be 1st, 2nd, 3rd,..., 9th?

How is it that 1 <= n <= 5 when there are 9 racers? Do racers 6 through 9 just get 0 points?

Yes, a racer gets points only when he/she ranks 1 - 5.

Break down the question to get a handle on it:
For a certain race, 3 teams were allowed to enter 3 members each.
This means 9 racers.

A team earned 6-n points whenever one of its members finished in nth place, where 1<= n<=5, there were no ties, or withdraw.
Since n varies from 1 to 5, only when a member finishes in one of those positions, does he score something. That something is 6 - n. So person who finishes first, gets 5 points, person who finishes 2nd gets 4 points and so on till the person who finishes 5th gets 1 point. So in all, 5+4+3+2+1 = 15 points were given

If no team earned more than 6 points, what is the least possible score a team could have owned?
No team got more than 6 points. We have to find the minimum score of a team. Since the total is 15 and one score has to be minimized, we should try to maximize the other two scores. Maximum score is 6 so other two teams get 6 points each maximum (e.g. One team gets 5 + 1, another gets 4+2). Then the third team will get a minimum score of 15 - 2*6 = 3

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caiyun

finishing place = 1, 2, 3, 4, and 5.

possible points:
1st place = 6 - 1 = 5
2nd place = 6 - 2 = 4
3rd place = 6 - 3 = 3
4th place = 6 - 4 = 2
5th place = 6 - 5 = 1

so total points = 5+4+3+2+1 = 15
If we assign maximum scores to 2 teams, then the remaining scores is for the remaining team.
max scores first 2 teams can have = 6 (6+1) + 6 (4+2) = 12
so the remaining team must have = 3 scores

so got D.

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should be 3 points.

the teams scored 6 - n points for each prize won.

this results in 5 points for first, 4 for second, etc.

The total number of points given for the race is 5 + 4 + 3 + 2 + 1, or 15 points awarded to all 3 teams combined.

since no team was awarded more than 6, then to get the least possible score, assume two teams earned 6 points.

15 - 2 (6) = 3

there are only 3 more points to be awarded, so the least possible score for the losing team would be 3.

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OK so total runners is 9..but only the first 5 actually get you any points...right

so lets say teams are T1, T2 and T3

T1 gets 1st spot so (6-1)=5, now we are told that no one scores over 6..so then assume that 2nd runner of T1 gets 6-5=1 point so in all T1 has 6 points..the 3rd runner was below 5th so he doesnt carry in points.anyway..

T2 gets 2nd spot so they 4 points from one of their runners, then their other runner gets 4th spot, and thus T2 also get 6 points..

T3 is left with only one runner who is at 3rd spot..so they get 3 points..

D it is

caiyun

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aniketb

Minimum possible score which team can earn through each member is 1. There are 3 members in each team. So, minimum possible score for a team is 1+1+1=3.

thats how i thought it to be. Experts please let me know if its correct.

From what I understand from your explanation, I don't think your method is correct.

Note that only one person will get a score of 1. There are 3 teams with 3 athletes each so there are a total of 9 athletes competing in the race.
The first 5 positions, will be given points 5, 4, 3, 2 and 1. So theoretically, a team could gather a max of 5+4+3 = 12 points if its three athletes get positions 1st, 2nd and 3rd. Also, theoretically, a team could get 0 points if its 3 athletes occupy the last 3 positions!

There are a total of 5+4+3+2+1 = 15 points up for grabs among the 9 athletes.

But you are given that the max points a team got was 6. Say, its two athletes got ranks 1st and 5th and hence scored 5 and 1 respectively.
To give minimum points to one team, we need to give max points to the other team too i.e. 6 (say, its two athletes got ranks 2nd and 4th).
So out of a total of 15 points, 6 each are allotted to two teams leaving you with 3 points for the third team (one of its athletes came in 3rd)

That is the reason 3 is the minimum points a team could get.

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Plz see below.

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This wording of this problem is still misleading because how can Team 1 and 2 both get 6 points total if it says there were no ties?

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shriyasp

This wording of this problem is still misleading because how can Team 1 and 2 both get 6 points total if it says there were no ties?

"No ties" means that there were no ties between members/runners not between teams.

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Total scores:
6-1 = 5
6-2 = 4
6-3 = 3
6-4 = 2
6-5 = 1
------------
Total=15
------------

Now, no team can get more than 6 points, and we need to find the least possible points that a team can have. Thus, lets give maximum points to 2 team so that the remaining one team will have least points.
Max points per team = 6
for 2 teams, max total points = 2X6 = 12.
Points remaining for team 3 = 15-12 = 3.

Answer D

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I3igDmsu

Responding to a pm:

Quote:

may i ask you a question about this problem: would it be correct to assume that the least nr of points that team could possibly earn in a race is 1 point (disregarding the case when no racer from a team scores any points), and so much as a result of each race, which yields 3*1=3 points?

I do understand your reasoning, but mine is a bit more straightforward, and I wonder if I am lucky to hit the correct answer with wrong assumption, or is it a valid reasoning?

No, this reasoning is not correct.

Look, there is only ONE race. The rankings in that race are the only points the racers get. The reason some points get added up is that the racers belong to a team.
Team A has three racers: A1, A2, A3.
Similarly, team B and team C.

If A1 gets 4 points and A3 gets 2 points, team A gets 4 + 2 = 6 points.

No team got more than 6 points. So say team A got 6 points and team B got 6 points. This means team C must have got 3 points.
That is the reason 3 points is the minimum any team would have got.

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I3igDmsu

I am clear with the approach, but have just one doubt. As it is mentioned in the questions that there were no ties and we are allocating 6 points to two team, won't that be considered as a tie?

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I3igDmsu

I am clear with the approach, but have just one doubt. As it is mentioned in the questions that there were no ties and we are allocating 6 points to two team, won't that be considered as a tie?

Each team has provided 3 runners. No tie means no two runners get the same position (and hence the same number of points out of 1/2/3/4/5). 2 teams get 6 points which would be split as 1+5 and 2+4 which means 2 of their runners got points for them.

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Hi All,

This is essentially a "limit" question; to figure out the LEAST number of points that any team could score, we have to MAXIMIZE what the other 2 teams scored.

Since a team scores (6-N) points for each racer who finishes in Nth place (1 <= N <= 5). This means that ONLY 5 racers get points:
1st = 5 points
2nd = 4 points
3rd = 3 points
4th = 2 points
5th = 1 point
Total = 15 possible points

We're told that none of the 3 teams scored more than 6 points.

IF we can find a way for two of the teams to score 6 points each, then the third team would have the remaining 3 points. There IS a way for that to occur (one team finishes 1st and 5th, one team finishes 2nd and 4th).

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A detailed video explanation can be found here:
https://www.youtube.com/watch?v=sJX8a4XlkL8

The person who finishes 1st gets 5 points, the person who finishes 2nd gets 4 points, etc., etc... the 5th place finisher gets 1 point, and no other runner recieves any points.

When you see a math question that asks about minimums, immediately think about maximums, as well (and vice versa).

The total number of poins is 5 + 4 + 3 + 2 + 1 = 15

We're told that the maximum score for one team is 6 points. The minumum score would equal 15 minus the scores of TWO teams that earn a maximum score. The minimum = 15 - 6 - 6 = 3

Its important to check and make sure that two teams can indeed get six points. If you line up 1, 2, 3, 4, 5 you'll see that runners for one team could place 1st and 5th, runners for a second team could place 2nd and 4th, and a third team could get three points by having a runner who places 3rd. Answer D checks out.

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VeritasKarishma

I3igDmsu

I arrived at the answer very easily... please let me know if my approach is right.

Question asked: ' least possible score a team could have earned'... so it is asking what a single team can earn and given that 3 members in a team is participating.

condition: in worst condition one can score --> 6-5=1
so 3 persons in a team.. 3*1=3

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Hi venkateshraviraj,

The prompt tells us that there are 3 teams with 3 players each - so up to 9 players could enter a race. However, ONLY the FIRST 5 to finish the race scored points. The amount of points rewarded is based on the given formula:

A team scores (6-N) points for each racer who finishes in Nth place (1 <= N <= 5).

1st = 5 points
2nd = 4 points
3rd = 3 points
4th = 2 points
5th = 1 point
6th - 9th place = 0 points

Total = 15 possible points

Thus, it's NOT correct to assume that the least points an individual could receive is 1... it's actually 0. If we had no other information to work with, then the least points that a team could earn would be 0 (if all 3 members finished in 6th - 9th place).

HOWEVER, we're also told that none of the 3 teams scored more than 6 points. There IS a way for two of the teams to score 6 points each (re: one team finishes 1st and 5th, one team finishes 2nd and 4th). Thus, the remaining 3 points (for 3rd place) would go to the final team - and the minimum number of points that could be earned would be 3.

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Hello,
Since the problem clearly states that there were no ties, no two teams could get equal points. Hence the solution must be:
-> team A -6 points
-> team B- 5 points
Hence, team C-> 15-6-5=4 points Answer.

Please correct me if I am wrong.

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