Two Hungarian teams, Bolyai TC and Eötvös TK, have made it to the Round of 16 of the UEFA Champions League. What is the probability that they will play against each other? (The competition follows a straight knockout format; in a match between two teams, one progresses while the other is eliminated.)
If they play according to a direct elimination system, first they form pairs from the teams (8 in total), and these pairs play against each other. From the 8 advancing teams, they again form pairs, etc. (The diagram shows the arrangement of matches for 8 starting teams.)

The 16 teams play a total of $8+4+2+1=15$ matches, and any two teams play either 1 or 0 matches against each other.

The pairs are formed randomly from the 16 teams. However, this can also be understood as the pairs and matches being given (i.e., who advances is also given), and the teams are "drawn into" this diagram. From this perspective, we can think that from the 16 teams (among which the matches, who advances, etc., are given), we select the two, one of which is Bolyai TC, the other is Eötvös TK. We are curious whether these two teams will play against each other.

The selection can be made in $16 \cdot 15 / 2=120$ ways. Since there are 15 matches, and any two teams play at most once, there are 15 pairs of teams that play against each other, and there are no matches among the others.

The probability is thus: $\frac{\text { number of favorable cases }}{\text { total number of cases }}=\frac{15}{120}=\frac{1}{8}$. This gives us the probability.

The problem can be generalized to $2^{n}$ teams. In this case, a total of $2^{n-1}+2^{n-2}+\cdots+2+1=2^{n}-1$ matches are played (since one team is eliminated in each match). Two teams can be selected in $\binom{2^{n}}{2}=\frac{2^{n}\left(2^{n}-1\right)}{2}$ ways, so the desired probability is:

$$
\frac{2^{n}-1}{\frac{2^{2}\left(2^{n}-1\right)}{2}}=\frac{2}{2^{n}}=\frac{1}{2^{n-1}}
$$

Elöd Csirmaz (Fazekas M. Főv. Gyak. Gimn., 9th grade)

Note. It was nice that Máthé András and Kiss Gergely solved the exercise even more generally, and Máthé András writes: "Although it was not specified in the problem that it would be about football, we can assume that the Hungarian teams and the other teams do not play equally. Therefore, let the winning probability of the Hungarian team against any foreign team be $q . . . "$

![](https://cdn.mathpix.com/cropped/2024_05_02_61f7f4c7a7f77aa594a5g-1.jpg?height=198&width=515&top_left_y=1213&top_left_x=794)