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Problem 14 Easy

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?

  • A.

    \frac { 2 } { 9 }

  • B.

    \frac { 2 } { 7 }

  • C.

    \frac { 9 } { 2 8 }

  • D.

    \frac { 1 } { 3 }

  • E.

    \frac { 3 } { 8 }

Answer:C
Solution

Total number of assigning bands:

\binom{9}{3\ 3\ 3} = \frac{9!}{3! \cdot 3! \cdot 3!}

For the three tallest, the number of ways to assign different bands is 3!, and for the remaining 6 players:

\binom{6}{2\ 2\ 2} = \frac{6!}{2! \cdot 2! \cdot 2!}

Thus, the probability:

= \frac{3! \cdot \frac{6!}{2! \cdot 2! \cdot 2!}}{\frac{9!}{3! \cdot 3! \cdot 3!}} = \frac{9}{28}

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