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.

calculate how many different dog families are possible.
(type a whole number.)

Explanation:

Identify the counting scenario

We need to choose 6 dogs out of 8 different dogs. Since we are selecting a group of dogs to form a "dog family," the order in which we select the dogs does not matter. Additionally, we cannot select the same dog more than once (no repetition).

Using the Combinations knowledge point, we determine that combinations are appropriate because the order of selection does not matter and items cannot be repeated.

Determine the correct option

We evaluate the given choices:

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

Calculate the number of families

Using the Combinations formula to find the number of ways to choose 6 dogs from 8:

$$ C(8, 6) = \frac{8!}{6!(8-6)!} = \frac{8 \times 7}{2 \times 1} = 28 $$

Answer:

Question 1

• 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.

Question 2

There are <blank>28</blank> different dog families possible.