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In a tournament with $7$ teams, each team plays one match with every other team. For each match, the team earns two points if it wins, one point if it ties, and no points if it loses. At the end of all matches, the teams are ordered in the descending order of their total points (the order among the teams with the same total are determined by a whimsical tournament referee). The first three teams in this ordering are then chosen to play in the next round. What is the minimum total number of points a team must earn in order to be guaranteed a place in the next round?

1. $13$
2. $12$
3. $11$
4. $10$
5. $9$

I think possible with 9 only.
no 9 will not guarantee!!!

Let the $7$ Teams be $A,B,C,D,E,F,G$ and so each team plays total $6$ matches.

Suppose, Team $A$ wins over $E,F,G$ and draws with $B,C,D$ hence total points scored by Team A $= 9$ points

Now, Team $B$ wins over $E,F,G$ and draws with $A,C,D$ hence total points scored by Team B $= 9$ points

Similarly, happens for next two teams $C$ and $D$ .

Hence, Finalized scores are $\Rightarrow$

A = 9
B = 9
C = 9
D = 9
E = ? (Less than or equal to 4)
F = ? ("...")
G = ? ("...")

Given that the order among the teams with the same total are determined by a whimsical tournament referee.

So, He/She can order the top $3$ teams like $ABC,ABD,BCD,ACD,\ldots$

But, Question says " team must earn in order to be guaranteed a place in the next round "

Hence, Not to depend on that whimsical referee, the minimum total number of points a team must earn in order to be guaranteed a place in the next round = $9+1 = 10$ points

Correct Answer: $D$

by

So only 2 teams have 10 point right ??
why  u considered A has won with 3 teams and tie with 3 teams?

i mean why not u took the case where A won with all the teams....(may be because we have to tell the min points required?)

but i am confused why 3 win & 3 tie?

why not 4 win and 2 tie...or any other case?

I think 3 teams can have 10 points.

$\underbrace{T_1 , T_2, T_3}_\text{Tie}, \underbrace{T_4,T_5,T_6,T_7}_\text {loses}$

T1,T2 and T3 will win against 4,5,6th and 7th and will gain 2*4 = 8 points, and it will be a tie between themselves;  2 points for that. Total = 10 points.

If someone wants to get into the next round he/she must secure at least 3rd position. That means that a team must earn as many points as the $3$rd team does to keep alive the hope of going into the next round. (sometimes +1 , we will get to that later.)

We assume that these winners are in the order $A\rightarrow B\rightarrow C$.

Now we will try to increase the points of team $C$ such that $\text{points}(A,B) \geq C$ and $C$ also maintain $3$rd position.

Consider teams $A,B,C$

There can be a situation when winners $A,B,C$ all three team beat $D,E,F,G$ and play draw among them. Then $A,B,C$ will get $10$ points each.

• $\Rightarrow$ So, $3$rd team $C$ can get maximum $10$ points.

$C$ can not get $11$ points. Because in that case it has to beat one of the winners , and it will move to higher position but we need $C$ at 3rd posotion only.

What happens to $D$ ? the $4$th position holder ? He can get maximum $6$ after $\text{three}$ consecutive loss to $A,B \text{ and }C$ by beating $E,F,G$.

• $\Rightarrow$ So, If a team gets $10$ points , that team definitely get into the next round.
• Points of $C$ and $D$ are not equal in this case and we need not worry about referee.
by

1 comment

Well explained.Thanks @  Debashish Deka

suppose there are $7$ teams.

 Team A B C D E F G Points 0 0 0 0 0 0 0

$A$ wins all the matches but loses to $B$

 Team ${\color{Green} A}$ B C D E F G Points ${\color{Green}{10}}$ $2$ 0 0 0 0 0

now since $A$ has played with all the teams next $B$ plays with remaining teams(i.e. $C,D,E,F,G$) and wins all the matches but loses to $C$

 Team ${\color{Green} A}$ ${\color{Green} B}$ C D E F G Points ${\color{Green}{10}}$ ${\color{Green}{10}}$ $2$ 0 0 0 0

now since $A,B$ have played with all the teams next $C$ plays with remaining teams(i.e. $D,E,F,G$) and wins all the matches.

 Team ${\color{Green} A}$ ${\color{Green} B}$ ${\color{Green}{C}}$ D E F G Points ${\color{Green}{10}}$ ${\color{Green}{10}}$ ${\color{Green}{10}}$ 0 0 0 0

Now all the remaining teams cannot make a score $\geq 10$ so only $A,B,C$ can enter the next round.

And the score of each of $A,B,C =10$

$\therefore$ Option $D.$  $10$ is the correct answer.

by

In your example, even if C wins in all matches against D,E,F,G, it will still get 8 points at max.How do you explain 9 or 10 points of C?
Yes you are correct my example was wrong.Updated it now.
$\text{A B C | D E F G}$

Suppose $\text{E, F, G}$ have disqualified. So we have to consider the minimum points these teams can score. It will not be $0$ as $\text{E,F,G}$ play matches among themselves.

Minimum points $\text{E,F,G}$ can get is $2,2,2$ resp.($\text{E-F F-G E-G}$ ties)

The total number of points is $42$ {$21$ games- $\binom{7}{2}$}

So $42-6 =36$ points left.

This can be distributed among teams $\text{A,B,C,D}$ in:

$\text{9 9 9 9}$

$\text{10 10 8 8}$ and on...

$8$ and $9$ cannot guarantee a team to qualify to second round as more than one team scores the same. So the minimum points to qualify to second round is $10$.

1 comment

why you assumed only, E,F and G are dis qualified ?

why not D ?

if you are not disqualifying D, then try to not disqualifying E also, then only F and G are disqualified !

How these are taken care in your solution ?

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