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Mathematics-Online problems:

Interactive Problem 1038: Group Fixtures during the Group Stage Matches of a World Cup


A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

The left table shows the fixtures of a group during the group stage matches of a World Cup. There is also the table with the points finally obtained by these teams:

Day of match Fixture
1st Andorra : Botswana
1st Cyprus : Djibouti
2nd Andorra : Cyprus
2nd Botswana : Djibouti
3rd Andorra : Djibouti
3rd Botswana : Cyprus

        

Place Team Points
1st Botswana 7
2nd Andorra 5
3rd Cyprus 4
4th Djibouti 0

a)
Indicate for each of the six fixtures how many points both teams obtained (win = 3 points, draw = 1 point, defeat = 0 points).

b)
How many combinations $ m$ of scores of the six games are altogether possible, i.e. in how many ways can the table be filled in?

c)
How many different possibilities $ n$ exist to determine the fixtures for the three days of match?

Hint: The sequence of the games within one day and of the teams within one fixture is not being considered.

d)
Tick whether the following final tables are possible with adequate courses of games assumed.

Table I Table II Table III
Place Team Pts.
1st Andorra 8
2nd Botswana 5
3rd Djibouti 4
4th Cyprus 1
Place Team Pts.
1st Cyprus 6
2nd Djibouti 5
3rd Botswana 4
4th Andorra 3
Place Team Pts.
1st Djibouti 7
2nd Cyprus 4
3rd Andorra 3
4th Botswana 2



Answer:

a)
Day of match Fixture Points
1st Andorra : Botswana :
1st Cyprus : Djibouti :
2nd Andorra : Cyprus :
2nd Botswana : Djibouti :
3rd Andorra : Djibouti :
3rd Botswana : Cyprus :

b)  
$ m =$
   
c)  
$ n =$

d)
Table I : not specified possible not possible
Table II : not specified possible not possible
Table III : not specified possible not possible


   

(From: Day of Mathematics 2006)

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  automatically generated: 8/11/2017