Answer :
To determine which expression represents the probability that all three awards will go to a student from school B, let's break down each of the expressions and understand what they represent.
### Information Given:
- There are 10 students from school A.
- There are 12 students from school B.
- There are a total of [tex]\( 10 + 12 = 22 \)[/tex] students competing.
- We need to find the probability that the first, second, and third place awards will go to students from school B.
### Analyzing Each Expression:
1. [tex]\(\frac{12 P_3}{22 P_3}\)[/tex]:
- Explanation: This expression uses permutation notation: [tex]\( nP_r \)[/tex] represents the number of ways to choose and arrange [tex]\( r \)[/tex] objects out of [tex]\( n \)[/tex] possible objects.
- Numerator ([tex]\(12 P_3\)[/tex]): This is the number of ways to choose and arrange 3 students out of 12 students from school B.
- Denominator ([tex]\(22 P_3\)[/tex]): This is the number of ways to choose and arrange 3 students out of all 22 students.
- Conclusion: This expression calculates the probability correctly as the ratio of favorable outcomes (all students from school B) to the possible outcomes (all students competing).
2. [tex]\(\frac{{ }_{12} C _3}{{ }_{22} C _3}\)[/tex]:
- Explanation: This expression uses combination notation: [tex]\( nC_r \)[/tex] represents the number of ways to choose [tex]\( r \)[/tex] objects out of [tex]\( n \)[/tex] possible objects without considering the order.
- Numerator ([tex]\({ }_{12} C _3\)[/tex]): This calculates the ways to choose 3 students out of 12 from school B, regardless of order.
- Denominator ([tex]\({ }_{22} C _3\)[/tex]): This calculates the ways to choose 3 students out of 22, regardless of order.
- Conclusion: This expression considers combinations without order, which does not correctly account for awarding the places (since order matters).
3. [tex]\(\frac{22^{P_3}}{22 P_{12}}\)[/tex]:
- Explanation: This expression uses a notation that is not standard for permutations or combinations and does not seem to make logical sense in the context.
- Conclusion: This expression is not correct.
4. [tex]\(\frac{{ }_{22} C_3}{{ }_{22} C_{12}}\)[/tex]:
- Explanation: This expression is not standard and does not logically relate to the problem at hand. The combination notation seems misplaced.
- Conclusion: This expression is not correct.
### Conclusion:
The correct expression representing the probability that all three awards will go to a student from school B is:
[tex]\[ \frac{12 P_3}{22 P_3} \][/tex]
This aligns with the analysis and calculation previously confirmed.
### Information Given:
- There are 10 students from school A.
- There are 12 students from school B.
- There are a total of [tex]\( 10 + 12 = 22 \)[/tex] students competing.
- We need to find the probability that the first, second, and third place awards will go to students from school B.
### Analyzing Each Expression:
1. [tex]\(\frac{12 P_3}{22 P_3}\)[/tex]:
- Explanation: This expression uses permutation notation: [tex]\( nP_r \)[/tex] represents the number of ways to choose and arrange [tex]\( r \)[/tex] objects out of [tex]\( n \)[/tex] possible objects.
- Numerator ([tex]\(12 P_3\)[/tex]): This is the number of ways to choose and arrange 3 students out of 12 students from school B.
- Denominator ([tex]\(22 P_3\)[/tex]): This is the number of ways to choose and arrange 3 students out of all 22 students.
- Conclusion: This expression calculates the probability correctly as the ratio of favorable outcomes (all students from school B) to the possible outcomes (all students competing).
2. [tex]\(\frac{{ }_{12} C _3}{{ }_{22} C _3}\)[/tex]:
- Explanation: This expression uses combination notation: [tex]\( nC_r \)[/tex] represents the number of ways to choose [tex]\( r \)[/tex] objects out of [tex]\( n \)[/tex] possible objects without considering the order.
- Numerator ([tex]\({ }_{12} C _3\)[/tex]): This calculates the ways to choose 3 students out of 12 from school B, regardless of order.
- Denominator ([tex]\({ }_{22} C _3\)[/tex]): This calculates the ways to choose 3 students out of 22, regardless of order.
- Conclusion: This expression considers combinations without order, which does not correctly account for awarding the places (since order matters).
3. [tex]\(\frac{22^{P_3}}{22 P_{12}}\)[/tex]:
- Explanation: This expression uses a notation that is not standard for permutations or combinations and does not seem to make logical sense in the context.
- Conclusion: This expression is not correct.
4. [tex]\(\frac{{ }_{22} C_3}{{ }_{22} C_{12}}\)[/tex]:
- Explanation: This expression is not standard and does not logically relate to the problem at hand. The combination notation seems misplaced.
- Conclusion: This expression is not correct.
### Conclusion:
The correct expression representing the probability that all three awards will go to a student from school B is:
[tex]\[ \frac{12 P_3}{22 P_3} \][/tex]
This aligns with the analysis and calculation previously confirmed.
