Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
The number of permutations of [tex]\( n \)[/tex] different objects can be described as the number of ways to arrange [tex]\( n \)[/tex] distinct items in a specific order. This count is commonly denoted as [tex]\( n! \)[/tex] (read as "n factorial").
Detailed Explanation:
1. Definition of Permutations:
- A permutation refers to an arrangement of objects in a specific order. For [tex]\( n \)[/tex] distinct objects, the number of permutations represents the total number of possible ways to order these objects.
2. Factorial Notation ([tex]\( n! \)[/tex]):
- The factorial of a non-negative integer [tex]\( n \)[/tex], denoted by [tex]\( n! \)[/tex], is the product of all positive integers less than or equal to [tex]\( n \)[/tex].
- Mathematically, [tex]\( n! \)[/tex] is defined as:
[tex]\[ n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 \][/tex]
3. Calculating Permutations:
- When calculating the permutations of [tex]\( n \)[/tex] different objects, we start with [tex]\( n \)[/tex] choices for the first position.
- Once the first position is filled, we have [tex]\( n-1 \)[/tex] choices for the second position.
- This process continues, reducing the number of available choices by one each time, until we fill the last position, which has exactly 1 choice left.
- Thus, the number of permutations is the product of all these choices, which precisely follows the definition of [tex]\( n! \)[/tex].
4. Example:
- For [tex]\( n = 3 \)[/tex], the permutations are calculated as:
[tex]\[ 3! = 3 \times 2 \times 1 = 6 \][/tex]
The different permutations of 3 distinct objects (say A, B, and C) are: ABC, ACB, BAC, BCA, CAB, and CBA, totaling 6 permutations. This aligns with the factorial calculation.
Given the definition and calculation above, the expression:
[tex]\[ n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 \][/tex]
is indeed correct in describing the number of permutations of [tex]\( n \)[/tex] different objects.
Therefore, the correct answer is: True.
Detailed Explanation:
1. Definition of Permutations:
- A permutation refers to an arrangement of objects in a specific order. For [tex]\( n \)[/tex] distinct objects, the number of permutations represents the total number of possible ways to order these objects.
2. Factorial Notation ([tex]\( n! \)[/tex]):
- The factorial of a non-negative integer [tex]\( n \)[/tex], denoted by [tex]\( n! \)[/tex], is the product of all positive integers less than or equal to [tex]\( n \)[/tex].
- Mathematically, [tex]\( n! \)[/tex] is defined as:
[tex]\[ n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 \][/tex]
3. Calculating Permutations:
- When calculating the permutations of [tex]\( n \)[/tex] different objects, we start with [tex]\( n \)[/tex] choices for the first position.
- Once the first position is filled, we have [tex]\( n-1 \)[/tex] choices for the second position.
- This process continues, reducing the number of available choices by one each time, until we fill the last position, which has exactly 1 choice left.
- Thus, the number of permutations is the product of all these choices, which precisely follows the definition of [tex]\( n! \)[/tex].
4. Example:
- For [tex]\( n = 3 \)[/tex], the permutations are calculated as:
[tex]\[ 3! = 3 \times 2 \times 1 = 6 \][/tex]
The different permutations of 3 distinct objects (say A, B, and C) are: ABC, ACB, BAC, BCA, CAB, and CBA, totaling 6 permutations. This aligns with the factorial calculation.
Given the definition and calculation above, the expression:
[tex]\[ n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 \][/tex]
is indeed correct in describing the number of permutations of [tex]\( n \)[/tex] different objects.
Therefore, the correct answer is: True.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
