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Combinatorics Difficulty 4.4 AIME Find the answer United States

Problem:

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?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

There are $\binom{128}{2} = 127 \cdot 64$ pairs of teams. In each tournament, $127$ of these pairs play.

By symmetry, the answer is $\frac{127}{127 \cdot 64} = \frac{1}{64}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.