To determine the correct formula for calculating the number of permutations of [tex]\( r \)[/tex] objects selected from [tex]\( n \)[/tex] possible objects, let's go step by step and examine the choices given.
Option 1: [tex]\( \frac{n!}{r!} \)[/tex]
This formula represents the number of permutations of [tex]\( n \)[/tex] objects taken [tex]\( n \)[/tex] at a time. It does not account for selecting [tex]\( r \)[/tex] objects from [tex]\( n \)[/tex].
Option 2: [tex]\( n \times r \)[/tex]
This formula is not appropriate for permutations. It simply represents a multiplication and doesn't properly account for the arrangement of [tex]\( r \)[/tex] objects from [tex]\( n \)[/tex] objects.
Option 3: [tex]\( \frac{n!}{(n-r)!} \)[/tex]
This is the well-known formula for permutations. It calculates the number of ways to arrange [tex]\( r \)[/tex] objects out of [tex]\( n \)[/tex] distinct objects. The numerator [tex]\( n! \)[/tex] (n factorial) accounts for the total arrangements of all [tex]\( n \)[/tex] objects, while the denominator [tex]\( (n-r)! \)[/tex] (factorial of the difference) eliminates the arrangements that only differ by the positions of the remaining [tex]\( n-r \)[/tex] objects that are not chosen.
Option 4: [tex]\( \frac{n!}{r(n-r)!} \)[/tex]
This formula is not correct for permutations; it resembles the formula for combinations, but even then, there's an error. For combinations, the correct formula would be [tex]\( \frac{n!}{r!(n-r)!} \)[/tex], which counts the ways to choose [tex]\( r \)[/tex] objects from [tex]\( n \)[/tex] without considering the order of arrangement.
After careful analysis, the correct formula for calculating the number of permutations of [tex]\( r \)[/tex] objects selected from [tex]\( n \)[/tex] possible objects is:
[tex]\[ {}_n P_r = \frac{n!}{(n-r)!} \][/tex]
Thus, the third option is the correct answer.
