Question

answer the following question using arrangements with repetition, permutations, or combinations. be sure to explain why the particular counting technique applies to the problem.

a dog shelter is giving away 8 different dogs, but you have room for only 6 of them. how many different dog families could you have?

determine the appropriate counting technique. choose the correct answer below.

a. combinations should be used because no item may be selected more than once and the order does not matter.
b. permutations should be used because no item may be selected more than once and the order matters.
c. arrangements with repetition should be used because we make selections from a group of choices.
d. permutations should be used because we make selections from a group of choices.
e. arrangements with repetition should be used because no item may be selected more than once and the order matters.

Explanation:

Analyze the selection criteria

We need to choose 6 dogs out of 8 different dogs. Since each dog is unique and cannot be chosen more than once, repetition is not allowed. Additionally, the group of 6 dogs forms a "family" where the order in which they are selected does not change the resulting group. Therefore, order does not matter.

Identify the counting technique

Using the Combinations concept, we select items without replacement where order is irrelevant. Using the Permutations concept, order would matter. Using the Arrangements with Repetitions concept, items could be selected multiple times. Thus, combinations is the correct technique.

Match with the options

Option A states: "Combinations should be used because no item may be selected more than once and the order does not matter." This perfectly matches our analysis.

Answer:

• A. Combinations should be used because no item may be selected more than once and the order does not matter. (Correct answer)
• B. Permutations should be used because no item may be selected more than once and the order matters.
• C. Arrangements with repetition should be used because we make selections from a group of choices.
• D. Permutations should be used because we make selections from a group of choices.
• E. Arrangements with repetition should be used because no item may be selected more than once and the order matters.