Question

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?

$\dfrac{1}{8}$

$\dfrac{5}{36}$

$\dfrac{14}{45}$

$\dfrac{25}{63}$

$\dfrac{1}{2}$

Solution:

Assume that Harold has already picked his $5$ books. Of these five books, there are $\binom{5}{2}$ ways that Betty can have picked exactly two of the same books as Harold, and $\binom{5}{3}$ ways that Betty can choose her other three books from the $5$ books not on Harold's list.

As such, there are $\binom{5}{2}\binom{5}{3}=100$ ways for Betty to choose her books such that she chooses exactly two books on Harold's list and three books not on Harold's list.

Therefore, as there are $\binom{10}{5}=252$ ways that Betty can choose her books arbitrarily, and $100$ of those choices satisfy the above conditions, the probability that they have exactly two books in common is: $\dfrac{100}{252} = \dfrac{25}{63}$

Thus, D is the correct answer.

2020 AMC 10B Problem 11

Solution:

Problem 11 in Other Years