Problem 14
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Nine athletes, no two of whom are the same height, try out for the basketball team. One at a time, they draw a wristband at random, without replacement, from a bag containing $3$ blue bands, $3$ red bands, and $3$ green bands. They are divided into a blue group, a red group, and a green group. The tallest member of each group is named the group captain. What is the probability that the group captains are the three tallest athletes?
$$\textbf{(A) } \frac{2}{9} \qquad \textbf{(B) } \frac {2}{7} \qquad \textbf{(C) } \frac{9}{28}\qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{3}{8}$$
Difficulty: 32/100 - A straightforward and intuitive probability problem.
Core Concepts: probabilities, constructive counting
Challenges: Not being able to tackle the without-replacement condition of the wristbands.
Techniques: Applying a constructive probability method where the probability of a different colored wristband is determined for each player.
Error-prone Steps: Forgetting the wristbands are selected without replacement.
Ideal Time:
Experienced: ≤ 1 min
Intermediate: ≤ 2-3 min
Beginner: ≤ 4 min