Answer :
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.
