AMC10 2020 B

AMC10 2020 B · Q11

AMC10 2020 B · Q11. It mainly tests Combinations, Probability (basic).

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?

Carr 女士要求她的学生从阅读列表中的 10 本书中阅读任意 5 本。Harold 随机从这个列表中选了 5 本书，Betty 也这样做。他们两人恰好都选了恰好 2 本相同书籍的概率是多少？

(A) \frac{1}{8} \frac{1}{8}

(B) \frac{5}{36} \frac{5}{36}

(D) \frac{25}{63} \frac{25}{63}

(E) \frac{1}{2} \frac{1}{2}

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Once Harold has selected his books, there are $\binom{5}{2}$ ways for Betty to choose 2 of Harold’s books and $\binom{5}{3}$ ways for her to choose 3 books that were not selected by Harold. There are $\binom{10}{5}$ ways in all for Betty to select her books. Thus the required probability is \[ \frac{\binom{5}{2}\cdot\binom{5}{3}}{\binom{10}{5}}=\frac{25}{63}. \]

答案（D）：一旦 Harold 选好了他的书，Betty 选 Harold 的书中 2 本有 $\binom{5}{2}$ 种方式，另从 Harold 未选的书中选 3 本有 $\binom{5}{3}$ 种方式。Betty 选 5 本书一共有 $\binom{10}{5}$ 种方式。因此所求概率为 \[ \frac{\binom{5}{2}\cdot\binom{5}{3}}{\binom{10}{5}}=\frac{25}{63}. \]

Topics