To solve the problem of finding the probability that all three awards will go to students from school B, we need to use the concept of permutations since the order in which the awards are given matters (first, second, third place).
First, let's define the variables:
- There are 12 students from school B.
- There are a total of 22 students (10 from school A + 12 from school B).
We need to find the probability that all three awards will go to students from school B.
### Step-by-Step Solution:
1. Calculate the number of ways to choose 3 students from school B to win the awards:
The number of ways to arrange 3 out of 12 students is given by the permutation of 12 students taken 3 at a time, denoted as [tex]\( 12P_3 \)[/tex].
2. Calculate the total number of ways to choose 3 students from the 22 students to win the awards:
The number of ways to arrange 3 out of 22 students is given by the permutation of 22 students taken 3 at a time, denoted as [tex]\( 22P_3 \)[/tex].
3. Determine the probability:
The probability is found by taking the ratio of the favorable outcomes to the total possible outcomes. This is represented as:
[tex]\[
\frac{12P_3}{22P_3}
\][/tex]
Therefore, the expression that represents the probability that all three awards will go to a student from school B is:
[tex]\[
\boxed{\frac{12P_3}{22P_3}}
\][/tex]
When calculated, this probability equals approximately [tex]\( 0.14285714285714285 \)[/tex] or roughly [tex]\( 1/7 \)[/tex].
