2

Part of 2015 AIME Problems

Problems(2)

Delegates to an Economic Conferece

Source: 2015 AIME I Problem 2

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

\tfrac{m}{n}

m

n

are relatively prime positive integers. Find

m+n

Fraction of Participants in a School

Source: 2015 AIME 2 Problem 2

In a new school

40

percent of the students are freshmen,

30

percent are sophomores,

20

percent are juniors, and

10

percent are seniors. All freshmen are required to take Latin, and

80

percent of the sophomores,

50

percent of the juniors, and

20

percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m+n