2020 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 11 Medium

Ms. Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this list, and Betty does the same. What is the probability that there are exactly 2 books that they both select?(2020 AMC 10B, Question #11)

  • A.

    \frac{1}{8}

  • B.

    \frac{5}{36}

  • C.

    \frac{14}{45}

  • D.

    \frac{25}{63}

  • E.

    \frac{1}{2}

Answer:D

We don't care about which books Harold selects. We just care that Betty picks 2 books from Harold's list and 3 that aren't on Harold's list. The total amount of combinations of books that Betty can select \left(\begin{array}{c}10 \\ 5\end{array}\right)=252 \left(\begin{array}{l}5 \\ 2\end{array}\right)=10 ways for Betty to choose 2 of the books that are on Harold's list. From the remàining 5 books that aren't on Hàrold's list, there \left(\begin{array}{l}5 \\ 3\end{array}\right)=10 \frac{10 \cdot 10}{252}=(\mathbf{D}) \frac{25}{63}

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