To determine the number of possible permutations of [tex]\( n \)[/tex] items taken [tex]\( r \)[/tex] at a time, we use the formula for permutations, which is given by:
[tex]\[ P(n, r) = \frac{n!}{(n-r)!} \][/tex]
Let's break this down:
- [tex]\( n! \)[/tex] is the factorial of [tex]\( n \)[/tex], which is the product of all positive integers up to [tex]\( n \)[/tex].
- [tex]\( (n-r)! \)[/tex] is the factorial of [tex]\( n-r \)[/tex], which is the product of all positive integers up to [tex]\( n-r \)[/tex].
This formula represents the number of ways to arrange [tex]\( r \)[/tex] items out of [tex]\( n \)[/tex] items in a specific order.
From the options given:
1. [tex]\(\frac{n!}{(n-r)!r!}\)[/tex]
2. [tex]\(\frac{n!}{(n-r)!}\)[/tex]
3. [tex]\(\frac{(n-r)!}{n!}\)[/tex]
4. [tex]\(\frac{(n-r)!r!}{n!}\)[/tex]
The correct expression that represents the number of possible permutations of [tex]\( n \)[/tex] items taken [tex]\( r \)[/tex] at a time is:
[tex]\[\frac{n!}{(n-r)!}\][/tex]
Therefore, the correct option is:
[tex]\[\boxed{\frac{n!}{(n-r)!}}\][/tex]
