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 15 different dogs, but you have room for only 3 of them. how many different dog families could you have?

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

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

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

Explanation:

Determine the appropriate counting technique

Using the Combinations, Permutations, and Arrangements with Repetitions knowledge points

• We must choose 3 distinct dogs from a group of 15 different dogs.
• Since we cannot choose the same dog more than once, repetition is not allowed.
• The order in which we choose the dogs does not matter because they form the same "dog family" regardless of selection order.
• Therefore, combinations should be used because no item may be selected more than once and the order does not matter.

Calculate the number of possible dog families

Using the Combinations knowledge point

$$ C(15, 3) = \frac{15!}{3!(15-3)!} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = 455 $$

Answer:

Question 1

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

Question 2

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