Problem
When you calculate the number of permutations of $n$ distinct objects taken $r$ at a time, what are you counting? Choose the correct answer below. A. The number of distinguishable ordered arrangements of $n$ objects taken $r$ at a time. B. The number of arrangements without regard to order of $n$ objects taken $r$ at a time. C. The number of distingushable arrangements without regard to order of $n$ objects taken $r$ at a time. D. The number of ordered arrangements of $n$ objects taken $r$ at a time.
Solution
Step 1 ：The question is asking about the definition of permutations of $n$ distinct objects taken $r$ at a time. Permutations are about counting different ways in which a set of objects can be arranged where the order of arrangement matters.
Step 2 ：Therefore, the correct answer should be the one that mentions both 'distinguishable' and 'ordered' arrangements.
Step 3 ：Final Answer: \(\boxed{\text{A. The number of distinguishable ordered arrangements of } n \text{ objects taken } r \text{ at a time.}}\)
