Sure, let's walk through the solution step-by-step to understand which expression correctly represents the probability that all three awards will go to students from school B.
1. Identify the total number of students and the number of students in each school:
- Number of students from school A: [tex]\(10\)[/tex]
- Number of students from school B: [tex]\(12\)[/tex]
- Total number of students: [tex]\(10 + 12 = 22\)[/tex]
2. Determine the scenario we are interested in:
- We want to find the probability that all three awards (for the first, second, and third places) go to students from school B.
3. Understand the nature of the problem:
- This is a problem of permutations because the awards are distinct (first place is different from second place, etc.).
4. Calculate the number of favorable outcomes:
- We need to calculate the number of ways to award 3 distinct positions from the 12 students in school B.
- This calculation is represented by the permutation function [tex]\(P(12, 3)\)[/tex], which is the number of ways to choose and arrange 3 students out of 12.
- The number of favorable permutations is given by [tex]\( P(12, 3) \)[/tex].
5. Calculate the total number of possible outcomes:
- We now need to calculate the total number of ways to award 3 distinct positions from all 22 students (from both school A and school B).
- This calculation is represented by the permutation function [tex]\(P(22, 3)\)[/tex], which is the number of ways to choose and arrange 3 students out of 22.
- The total number of permutations is given by [tex]\( P(22, 3) \)[/tex].
6. Formulate the probability:
- The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.
- Hence, the probability that all three awards will go to students from school B is given by the expression:
[tex]\[
\frac{P(12, 3)}{P(22, 3)}
\][/tex]
Given the provided multiple-choice options:
[tex]\[
\frac{12 P_3}{22 P_3} \quad \text{(option A)}
\][/tex]
[tex]\[
\frac{_{12} C_3}{_{22} C_3} \quad \text{(option B)}
\][/tex]
[tex]\[
\frac{_{22} P_3}{_{22} P_{12}} \quad \text{(option C)}
\][/tex]
[tex]\[
\frac{_{22} C_3}{_{22} C_{12}} \quad \text{(option D)}
\][/tex]
Option (A):
[tex]\[
\frac{12 P_3}{22 P_3}
\][/tex]
is the correct choice because it matches the expression we derived for calculating the probability that all three awards will go to students from school B.
The resulting values from the calculations in the given problem confirm this:
- [tex]\(P(12, 3) = 1320\)[/tex]
- [tex]\(P(22, 3) = 9240\)[/tex]
- Probability = [tex]\(\frac{1320}{9240} = 0.14285714285714285\)[/tex]
Therefore, the correct expression representing the probability is:
[tex]\[
\frac{12 P_3}{22 P_3}
\][/tex]
