the track team gives awards for first, second, and third place runners. there are 10 students from school a…

the track team gives awards for first, second, and third place runners. there are 10 students from school a and 12 students from school b competing. which expression represents the probability that all three awards will go to a student from school b? (\frac{_{12}p_{3}}{_{22}p_{3}}) (\frac{_{12}c_{3}}{_{22}c_{3}}) (\frac{_{22}p_{3}}{_{22}p_{12}}) (\frac{_{22}c_{3}}{_{22}c_{12}})

the track team gives awards for first, second, and third place runners. there are 10 students from school a and 12 students from school b competing. which expression represents the probability that all three awards will go to a student from school b? (\frac{_{12}p_{3}}{_{22}p_{3}}) (\frac{_{12}c_{3}}{_{22}c_{3}}) (\frac{_{22}p_{3}}{_{22}p_{12}}) (\frac{_{22}c_{3}}{_{22}c_{12}})

Answer

Explanation:

Step1: Calculate total number of students

The total number of students is (10 + 12=22) students.

Step2: Understand the nature of the problem

Since first, second, and third - place awards are distinct (order matters), we use permutations.

Step3: Calculate number of ways to choose 3 students from school B

The number of ways to choose 3 students from 12 students of school B (where order matters) is (_{12}P_3=\frac{12!}{(12 - 3)!}=\frac{12!}{9!}=12\times11\times10).

Step4: Calculate number of ways to choose 3 students from all students

The number of ways to choose 3 students from 22 students (where order matters) is (_{22}P_3=\frac{22!}{(22 - 3)!}=\frac{22!}{19!}=22\times21\times20).

Step5: Calculate the probability

The probability that all three awards go to a student from school B is the ratio of the number of ways to choose 3 students from school B to the number of ways to choose 3 students from all students, which is (\frac{{12}P_3}{{22}P_3}).

Answer:

(\frac{{12}P_3}{{22}P_3})