To calculate the probability that all three awards will go to students from school B, we need to consider both the number of possible successful outcomes (where all three winners are from school B) and the total number of possible outcomes (where the winners can be from either school).
1. Calculating the number of successful outcomes:
We have 12 students from school B. The number of ways to arrange 3 winners out of these 12 students (i.e., permutations) is calculated as:
[tex]\[
^{12}P_3
\][/tex]
This represents the number of successful outcomes.
2. Calculating the total number of outcomes:
We have a total of 22 students (10 from school A and 12 from school B). The number of ways to arrange 3 winners out of these 22 students (i.e., permutations) is calculated as:
[tex]\[
^{22}P_3
\][/tex]
This represents the total number of outcomes.
3. Calculating the probability:
The probability that all three awards will go to students from school B is the ratio of successful outcomes to the total outcomes:
[tex]\[
\frac{^{12}P_3}{^{22}P_3}
\][/tex]
When we calculate these values and their ratios, we get the probability as follows:
- The number of successful outcomes [tex]\((^{12}P_3)\)[/tex] is 1320.
- The total number of outcomes [tex]\((^{22}P_3)\)[/tex] is 9240.
- So, the probability is:
[tex]\[
\frac{1320}{9240} = 0.14285714285714285
\][/tex]
Thus, the correct expression representing the probability is:
[tex]\[
\boxed{\frac{^{12}P_3}{^{22}P_3}}
\][/tex]
