HMMT-Feb 2017 guts, Problem 8

You have 128 teams in a single elimination tournament. The Engineers and the Crimson are two of these teams. Each of the 128 teams in the tournament is equally strong, so during each match, each team has an equal probability of winning. Now, the 128 teams are randomly put into the bracket. What is the probability that the Engineers play the Crimson sometime during the tournament?

Solution: There are $\left(\begin{array}{c}128 \\ 2\end{array}\right)=127 \cdot 64$ pairs of teams. In each tournament, 127 of these pairs play. By symmetry, the answer is $\frac{127}{127 \cdot 64}=\frac{1}{64}$. $\fbox{\frac{1}{64}}$.

Answer: \frac{1}{64}