Question

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}})

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:

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