Given there are 10 students from school A and 12 students from school B competing, making a total of 22 students. We are to determine the probability that all three awards will go to students from school B. Let's go through the process step-by-step to calculate this probability.
1. Total Students:
[tex]\[ 10 \text{ (from school A)} + 12 \text{ (from school B)} = 22 \text{ students} \][/tex]
2. Probability for the First Award:
The number of ways to choose the first award winner from school B is 12 out of the total 22 students.
[tex]\[ \frac{12}{22} \][/tex]
3. Probability for the Second Award:
After one student from school B has already won an award, there are now 11 students remaining from school B and a total of 21 students.
[tex]\[ \frac{11}{21} \][/tex]
4. Probability for the Third Award:
After two students from school B have already won awards, there are now 10 students remaining from school B and a total of 20 students.
[tex]\[ \frac{10}{20} \][/tex]
5. Combined Probability:
The overall probability that all three awards will go to students from school B is the product of these individual probabilities:
[tex]\[ \frac{12}{22} \times \frac{11}{21} \times \frac{10}{20} \][/tex]
From our calculations, let's find the option that matches this setup:
- The expression [tex]\(\frac{12 P_3}{22 P_3}\)[/tex] refers to the permutation probability of selecting students for the awards, which aligns with our calculated method. Therefore, this represents the probability correctly.
Thus, the correct expression to represent the probability that all three awards will go to a student from school B is:
[tex]\[\boxed{\frac{12 P_3}{22 P_3}}\][/tex]
