At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Discover detailed solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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]
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]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
