To determine the probability that all three awards will go to students from school B, we first need to understand the problem and the provided options.
### Step-by-Step Solution:
1. Identifying Key Components:
- There are 10 students from school A.
- There are 12 students from school B.
- The total number of students competing is [tex]\(10 + 12 = 22\)[/tex].
- We need to find the probability that all three awards (first, second, and third places) are given to students from school B.
2. Permutations Explanation:
- We utilize permutations because the order in which the students receive the awards matters (first, second, third).
- The permutation formula for selecting [tex]\(r\)[/tex] out of [tex]\(n\)[/tex] objects is given by:
[tex]\[
P(n, r) = \frac{n!}{(n-r)!}
\][/tex]
3. Permutations Calculation:
- Number of ways to distribute 3 awards among 12 students from school B can be calculated as:
[tex]\[
P(12, 3)
\][/tex]
- Number of ways to distribute 3 awards among all 22 students can be calculated as:
[tex]\[
P(22, 3)
\][/tex]
4. Probability Calculation:
- The probability that all three awards will go to students from school B is the ratio of the number of favorable outcomes (students from school B) to the number of possible outcomes (all students):
[tex]\[
\text{Probability} = \frac{P(12, 3)}{P(22, 3)}
\][/tex]
5. Final Answer:
- From the above explanation, the expression that correctly represents this probability is:
[tex]\[
\frac{P(12, 3)}{P(22, 3)}
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{\frac{12 P _3}{{ }_{22} P _3}}
\][/tex]
