Question

Из 10 учащихся кружка математики 3 необходимо послать на олимпиаду. Сколькими способами можно составить команду, если; 1)все они идут городскую олимпиаду 2) один идет в районную, второй в городскую, третий в областную олимпиаду?

Answer 1

С 10/3 = 10!/((10-3)!*3!) = 10*9*8/(1*2*3) = 120 (спос.)
А 10/3  = 10!/(10-3)! = 8*9*10 = 720 (спос.)

Answer 2

Question Analysis

The question asks for the number of ways to form a team from a group of 10 students in a math club, given two scenarios: 1) All students participate in the city-level math Olympiad. 2) One student participates in the district-level Olympiad, another in the city-level Olympiad, and a third in the regional-level Olympiad.

Scenario 1: All students participate in the city-level math Olympiad

To calculate the number of ways to form a team where all students participate in the city-level math Olympiad, we need to select 3 students from a group of 10. This can be calculated using the combination formula:

Number of ways = nCr(n, r) = n! / (r! * (n-r)!)

In this case, n = 10 (total number of students) and r = 3 (number of students to be selected).

Scenario 2: One student participates in the district-level Olympiad, another in the city-level Olympiad, and a third in the regional-level Olympiad

To calculate the number of ways to form a team where one student participates in the district-level Olympiad, another in the city-level Olympiad, and a third in the regional-level Olympiad, we need to select 3 students from a group of 10, with specific roles assigned to each student.

The first student can be selected in 10 ways (any of the 10 students can be chosen). The second student can be selected in 9 ways (since one student has already been chosen). The third student can be selected in 8 ways (since two students have already been chosen). Therefore, the total number of ways to form the team in this scenario is:

Number of ways = 10 * 9 * 8 = 720

Final Answer

To summarize: 1) If all students participate in the city-level math Olympiad, there are 120 ways to form a team. 2) If one student participates in the district-level Olympiad, another in the city-level Olympiad, and a third in the regional-level Olympiad, there are 720 ways to form a team.

Please let me know if anything is unclear or if you have any further questions!