2015 AIME I 第 2 题

Question

经济合作会议的九名代表中，有

2

名来自墨西哥的官员、

3

名来自加拿大的官员，以及

4

名来自美国的官员。在开幕会议期间，其中三名代表睡着了。假设这三名睡着的代表是随机确定的，恰有两名睡着者来自同一个国家的概率为

\frac{m}{n}

，其中

m

与

n

是互质的正整数。求

m + n

。

The nine delegates to the Economic Cooperation Conference include

2

officials from Mexico,

3

officials from Canada, and

4

officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is

\frac{m}{n},

where

m

and

n

are relatively prime positive integers. Find

m + n.

答案

答案：139

知识点：基本概率组合分类讨论

难度评级：2110

小提示：

所有

\binom{9}{3}

个睡着者三人组等可能；按提供这一对代表的国家来计数有利情况

All

\binom{9}{3}

sleeper triples are equally likely; count the favorable ones by which country supplies the pair

大提示：

从人数为

s

的国家恰选两人，意味着先有

\binom{s}{2}

种选法，再从另外

9 - s

人中选第三名睡着者

Exactly two from a country of size

s

means

\binom{s}{2}

choices for the pair times

9 - s

choices for the third sleeper

解答：

三名睡着者共有

\binom{9}{3} = 84

个等可能的集合。恰有两人来自同一国家时，一个国家提供恰好两人，第三人来自不同国家：这一对来自美国有

\binom{4}{2}(2 + 3) = 30

种；来自加拿大有

\binom{3}{2}(2 + 4) = 18

种；来自墨西哥有

\binom{2}{2}(3 + 4) = 7

种。

概率为

\frac{30 + 18 + 7}{84} = \frac{55}{84}

，已经是最简分数，所以

m + n = 55 + 84 = 139

。

There are

\binom{9}{3} = 84

equally likely sets of three sleepers. Exactly two sleepers come from the same country when one country supplies exactly two of them and the third sleeper comes from a different country:

\binom{4}{2}(2 + 3) = 30

ways with the pair from the United States,

\binom{3}{2}(2 + 4) = 18

with the pair from Canada, and

\binom{2}{2}(3 + 4) = 7

with the pair from Mexico.

The probability is

\frac{30 + 18 + 7}{84} = \frac{55}{84},

already in lowest terms, so

m + n = 55 + 84 = 139.

2015 AIME I 第 2 题

答案：139

难度评级：2110

小提示：

大提示：

解答：

其他年份的第 2 题