Let's solve this step-by-step to determine the expression representing the probability that all three awards will go to students from school B.
1. Total Number of Students:
There are 10 students from school A and 12 students from school B, making a total of 22 students.
2. Objective:
We need to find the probability that all three awards (first, second, and third place) will go to students from school B.
3. Concepts Used:
- Permutations (P): Denoted as [tex]\( nP_r \)[/tex], which represents the number of ways to arrange [tex]\( r \)[/tex] objects out of [tex]\( n \)[/tex] objects.
- Combinations (C): Denoted as [tex]\( nC_r \)[/tex], which represents the number of ways to choose [tex]\( r \)[/tex] objects out of [tex]\( n \)[/tex] objects without regard to the order.
4. Calculation of Desired Probability:
- Permutations:
- The number of ways to choose and arrange 3 students out of the 12 students from school B is [tex]\(\text{perm}(12, 3)\)[/tex].
- The number of ways to choose and arrange 3 students out of the total 22 students is [tex]\(\text{perm}(22, 3)\)[/tex].
- The probability is then the ratio of these two permutations:
[tex]\[
\frac{\text{perm}(12, 3)}{\text{perm}(22, 3)}
\][/tex]
5. Simplify the Expression:
[tex]\[
\frac{12 \times 11 \times 10}{22 \times 21 \times 20}
\][/tex]
However, we do not need to simplify it further for this step.
6. Finding the Correct Expression:
Looking at the provided options:
- [tex]\(\frac{{12}^{P_3}}{22^{P_3}}\)[/tex] (Not a standard notation)
- [tex]\(\frac{{12 C _3}}{22 C_ 3}\)[/tex] (Using combinations: incorrect)
- [tex]\(\frac{22 P_3}{22 P_{12}}\)[/tex] (Incorrect as it doesn't make sense contextually)
- [tex]\(\frac{{22 C_ 3}}{22 C_{12}}\)[/tex] (Incorrect use of combinations for different purposes)
The correct expression should represent the ratio of the number of permutations of choosing 3 students from 12 to the number of permutations of choosing 3 students from 22.
Thus, the correct expression is:
[tex]\[
\frac{\text{perm}(12, 3)}{\text{perm}(22, 3)}
\][/tex]
Given the answer derived above (the numerical result), this probability is:
[tex]\[
0.14285714285714285
\][/tex]
