World Cup 2014

Probability Level 2

The World Cup is coming! Now I have a problem associated with it. Read the question carefully.

The number of countries who join the World Cup is $32$ . They qualify from World Cup qualifiers.The competition is divided into 2 phases: The Group Stage and The Knockout Phase.

1. The Group Stage. The $32$ countries will be divided into $8$ groups, with each group containing $4$ teams each. The teams in the same group will compete with all of the remaining team. $3$ points are awarded for a win, $1$ point is awarded for a draw, and no point are awarded for a defeat. Thus the maximum number of points one team would get is $9$ and the minimum number of points one team would get is $0$ . When all teams play one another, only the 2 teams from each group will qualify to the Knockout Phase.

$2$ . The Knockout Phase . Winner from one group will compete with the runner-up from another group in a random manner. The Knockout Phase will be divided into 4 sub-phases as follow:

2.1 The Round of 16 The teams qualifying from the Group Stage will compete with another team. Only the winners will proceed to the next round.

2.2 The Quarter-Finals The teams qualifying from the Round of 16 will compete with another team. Only the winners will proceed to the next round.

2.3 The Semi- Finals The teams qualifying from the Quarter-Finals will compete with another team. The winners and the losers will proceed to the next round.

2.4 The Final The winning teams from the Semi-Finals will compete in the final decisive match to determine which team is the champion.

2.5 The Match For The Third Place The losing teams from the Semi-Finals will compete to decide who is the Third in the competition.

Robert is an insane World Cup fan. He wants to watch ALL of the matches. How many matches will he watch?

The answer is 64.

7 solutions

Michael Diao
May 19, 2014

As described above, there are two main stages to the game. We can separate them by how they are paired.

In Stage 1 (the Group Stage), we have $8$ groups of $4$ teams each. Within those groups, each group plays the same number. In order to have all teams within a group compete with each other, we must pick two from four. This simplifies to $\binom{4}{2} = \frac{4 \times 3}{2!} = 6.$ Since we have $8$ groups, we must multiply $6$ by $8$ to get $\displaystyle{48}$ games played in the first round.
Stage 2 is easier. Since there are $16$ teams left, and the game is single-elimination, we will have a total of $16-1=15.$ But wait! We aren't done yet! The last part of the problem states that there will be one more game played to determine $3^{rd}$ place. Our final sum is $48+16=$