Counting Methods for Choosing Objects

D

We need the general permutation formula, which gives the number of ways that k objects can be chosen from n items, with regard to order. The formula is n_P_k

= n! / (n-k)!

The counting problem you described involves selecting k objects from a total of n objects, and the order in which the objects are chosen matters. In such a situation, you need to use the permutation rule, which is represented by option D.

The permutation rule is used when you want to count the number of ways to arrange objects in a specific order. It is applicable when the order matters, and repetition is not allowed.

To understand the permutation rule, let's break it down step by step:

1. Number of Choices for the First Object: Since you are selecting objects in a specific order, for the first object, you have n choices. This means you can choose any of the n objects.

2. Number of Choices for the Second Object: After choosing the first object, there are (n-1) objects remaining. Therefore, for the second object, you have (n-1) choices.

3. Number of Choices for the Third Object: Following the same logic, for the third object, you will have (n-2) choices. This pattern continues until you have chosen k objects.

To determine the total number of arrangements, you multiply the number of choices at each step together. Therefore, the total number of arrangements or permutations can be calculated as:

Total Permutations = n * (n-1) * (n-2) * ... * (n - k + 1)

In this formula, n represents the total number of objects, and k represents the number of objects you want to select.

This formula gives you the correct method for counting permutations when the order matters and repetition is not allowed. It is specifically applicable to the problem you described, where you are choosing k objects from a total of n objects.

Hence, the correct answer to the counting problem you provided is option D: The permutation rule.