AMC 10 Weekly Practice Round 3

Complete problem set with solutions and individual problem pages

Problem 20 Hard

To assess students’ physical fitness, 3 items are chosen at random from a total of 7 events: 3 track and field events, 2 ball games, and 2 martial arts events. What is the probability that the chosen 3 items come from exactly two categories?

  • A.

    \dfrac{22}{35}

  • B.

    \dfrac{2}{7}

  • C.

    \dfrac{4}{7}

  • D.

    \dfrac{11}{35}

  • E.

    \dfrac{33}{35}

Answer:A

The total number of ways to choose 3 items from these 7 events is

\,_{7}C_{3} = 35.

 

The number of ways where all 3 items come from the same category is

\,_{3}C_{3} = 1.

 

The number of ways where the 3 items cover all three categories is

\,_{3}C_{1}\,_{2}C_{1}\,_{2}C_{1} = 12.

 

Therefore, the number of favorable cases (exactly two categories) is

35 - 1 - 12 = 22.

 

So the required probability is \dfrac{22}{35}.

 

Hence, the answer is \dfrac{22}{35}.

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