To determine which expression represents the number of possible permutations of [tex]\( n \)[/tex] items taken [tex]\( r \)[/tex] at a time, we need to understand what a permutation is.
A permutation of [tex]\( n \)[/tex] items taken [tex]\( r \)[/tex] at a time is an arrangement of [tex]\( r \)[/tex] items out of the [tex]\( n \)[/tex] available, where order matters. The formula for permutations is given by:
[tex]\[ P(n, r) = \frac{n!}{(n-r)!} \][/tex]
Here, [tex]\( n! \)[/tex] (n factorial) represents the product of all positive integers up to [tex]\( n \)[/tex], and [tex]\( (n-r)! \)[/tex] is the factorial of the difference between [tex]\( n \)[/tex] and [tex]\( r \)[/tex].
Now let's evaluate the given options to find the correct expression for the number of permutations:
1. [tex]\(\frac{n \mid}{(n-r)|r|}\)[/tex]
This is not a standard permutation formula.
2. [tex]\(\frac{n!}{(n-r)!}\)[/tex]
This matches our formula for permutations, [tex]\( P(n, r) \)[/tex].
3. [tex]\(\frac{(n-r)!}{n!}\)[/tex]
This is the inverse of the correct permutation formula.
4. [tex]\(\frac{(n-r)|r|}{n!}\)[/tex]
This expression does not match the standard permutation formula.
Given that the correct permutation formula is [tex]\(\frac{n!}{(n-r)!}\)[/tex], the answer is the second option:
[tex]\[ \boxed{2} \][/tex]
