S96-11

Question

In the Mundane Goblet competition, 6 teams compete in a "round robin" format: that is, each team plays every other team exactly once. A team gets 3 points for a win, 1 point for a tie (a draw), and 0 points for a loss. What is the difference between the maximum total points and the minimum total points that can be gained by all teams (added together) in the Mundane Goblet competition?

A. 15
B. 30
C. 45
D. 60
E. 75

Show Answer

Official Answer

A

A. 15
B. 30
C. 45
D. 60
E. 75

Bunuel · Answer

Official Solution:

In the Mundane Goblet competition, 6 teams compete in a "round robin" format: that is, each team plays every other team exactly once. A team gets 3 points for a win, 1 point for a tie (a draw), and 0 points for a loss. What is the difference between the maximum total points and the minimum total points that can be gained by all teams (added together) in the Mundane Goblet competition?

A. 15
B. 30
C. 45
D. 60
E. 75

Show Answer

Official Answer

A

A. 15
B. 30
C. 45
D. 60
E. 75

First, we should determine the number of games played in this competition. We can count them in at least 2 different ways:
 
(1) Brute force. Name the 6 teams A, B, C, D, E, and F. A plays each of the other teams once, so A plays 5 games. B also plays 5 games, but we've already counted 1 of those games (the game with A), so we have 4 "new" games. C also plays 5 games, but we've already counted 2 of those games (the games with A and with B), so we have 3 "new" games. Continuing, we get  5 + 4 + 3 + 2 + 1 = 15  games.
 
(2) Combinatorics. We have a pool of 6 teams, and we want to count how many different pairs of teams (to play a game) we can select, without caring about order. Using either the anagram method or the formula for combinations, we get  \frac{6!}{2!4!} = 15  games.
 
Now, to find the maximum and minimum total points earned by all teams in the competition, we should notice that if one team wins and the other team loses, then 3 points total are earned (3 for the win and 0 for the loss). On the other hand, if the game ends in a draw, then only 2 points total are earned (1 by each team). So the maximum total points are earned if every game ends in a win/loss, and the minimum total points are earned if every game ends in a draw.
 
 	ext{Maximum} = 3 	imes 15 = 45  points.
 
 	ext{Minimum} = 2 	imes 15 = 30  points.
 
The difference between the maximum and the minimum is therefore  45 - 30 = 15 .

Answer: A

shapla · Answer

I think this the explanation isn't clear enough, please elaborate.

chetan2u · Answer

Hi  shapla ,

let us count the number of games played first..
total 6 teams.. each team plays with each of the other 5 teams..
so total games is ways of choosing 2 teams out of 6 teams=6C2=6!/4!2!=15 games..

now lets see the max and min points in each game..
a) if there is a result as win/loss, total points earned=3(for the winning team ) + 0(for a losing team)=3..
b) if there is a tie = 1+ 1(1 each for both team)=2
therefore the difference is 3-2=1..

there are 15 games, so total difference can be 15*1=15..
A
hope it was helpful

shapla · Answer

thanks chetan...

sommya · Answer

Can the minimum not be zero? What if there's a team that has lost in every match?

Bunuel · Answer

We need the minimum total points that can be gained by  all  teams (added together). How can the total points gained by all teams be 0?

KTH · Answer

So here is how i approached this problem. I followed the steps below:

a) Create 6x6 matrix
b ) You eliminate the diagonal , now we have : 36-6=30 left
c) Since the teams play just one time with each other we need only 30/2=15 elements
d) 3*15( max points) - 1*2*15(min points and 2 is here because both teams will get points)= 15
So answer : 15

Ajstyles · Answer

will some one please explain how the minimum score is 30.

SS7 · Answer

Total number of games played =15 (6C2)

Total minimum score-> every match played was a draw (In every match 1 point was given to both the teams) so total points in that case= 15*2= 30

bawaitey · Answer

Lets go the literal way:

Total number of games: AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF = 15 games

Team A winnings: A vrs B = 3 points + A vrs C = 3 + A vrs D = 3 + A vrs E = 3 + A vrs F = 3. This gives a total of 15
Team B winnings: B vrs A = 0 points because already lost to A above + B vrs C = 3 + B vrs D = 3 + B vrs E = 3 + B vrs F = 3. This gives a total of 12
Repeating the same logic: Team C winnings will equal 9, Team D will equal 6, Team E 3 and F 0.

TOTAL WINNING = 15+12+9+6+3+0 = 45 points

Total Draws: 1 point for both sides in a game gives us 2 points per game multiplied by 15 games = 30 points

Difference = 45 -30 = 15. Answer is A

MariannaMagari · Answer

If a team has lost every match, another team has won 3 points in every match. Indeed, the case in which one team looses every match ( and by consequence another one won ) is the case with the maximum total.

akadiyan · Answer

I missed some finer details in the question and chose the wrong answer

I calculated the number of matches correctly as 15 matches and maximum points a team can gain is 45 points by winning all the matches.

On the other hand , when calculating the minimum points, i counted the points for a draw only once for a team and ended up with 15 points. I then calculated 45-15 = 30 points.

I missed to consider that both the team gets 1 point each during the draw, which leads to 30 points for minimum number of points for the team added together.

Correct answer is 45-30 = 15 points.

ManoramaS · Answer

@bunnel
why minimum point is not zero.In a team of two if one wins then another will loss.One team will have maximum point and another will have mimimum

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