Team score

showbu 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

IMO D

If a member of a team comes in first 5 then only team gets the point .

Total number of points allocated for first 5 positions is : 1+2+3+4+5 = 15

and condition given is no team has more than 6 points , if we assume that 2 teams gathered 6 points each then 3 points has to be earned by 3rd team.

and that is the least possible score that team can have.

In my opinion the answer to this question would be "D" - 3 points.

Here is my thought process:
It is stated in the problem that no team gets more than 6 points (team score <= 6 points). And since the problem calls for minimum score of any (one) team, we could arrive at the minimum by maximising the score of the other two teams.

Team one: Let us suppose that they scored exactly 6 points (the maximum possible). This means two players within this team may have been placed in first and fifth position (2nd and 4th position is also possible - but I would let this be the finishing position of members from team 2). The total points earned by the team would therefore be: (6 - 1) + (6 - 5) = 5 + 1 = 6.

Team Two: Let us suppose that they scored exactly 6 points (the maximum possible). Assuming that the players within this team finishes in 2nd and 4th place, we get total team score of (6 - 2)+(6 - 4) = 4 + 2 = 6.

Team Three: Now since places 1 and 5 are taken by team one and places 2 and 4 by team two, the only place that remains is position number 3 and this is where one member of team three gets placed. The total team score for team three should therefore be 6 - 3 = 3 points.