To determine which option does NOT meet the requirement of the Permutations Rule, we need to understand each of the listed choices and how they relate to the permutations formula [tex]\( {}_nP_r = \frac{n!}{(n-r)!} \)[/tex].
1. Option A: Order is taken into account (rearrangements of the same items are considered to be different).
- This is indeed true for permutations. The fundamental aspect of permutations is that the order in which items are arranged matters. Hence, different arrangements (or permutations) of the same set of items are considered different.
2. Option B: There are n different items available.
- This is also a requirement for permutations. To calculate permutations, you need to have a distinct set of [tex]\( n \)[/tex] items to choose from.
3. Option C: Exactly [tex]\( r \)[/tex] of the [tex]\( n \)[/tex] items are selected (without replacement).
- This is correct as well. Part of the permutations definition is choosing [tex]\( r \)[/tex] items out of [tex]\( n \)[/tex] without replacement, meaning once an item is chosen, it cannot be chosen again.
4. Option D: Order is not taken into account (rearrangements of the same items are considered to be the same).
- This statement is false regarding permutations because order not being taken into account aligns more with combinations, not permutations. In permutations, order does matter, and different orders of the same items are considered different permutations.
After evaluating each option, we can conclude that the statement that does NOT meet the requirement of the Permutations Rule is:
D. Order is not taken into account (rearrangements of the same items are considered to be the same).
This statement contradicts the fundamental property of permutations where the order does matter.
