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 city council with nine members must elect a four-person executive committee consisting of a mayor, deputy mayor, secretary, and treasurer. how many executive committees are possible?

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

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

Explanation:

Analyze the selection constraints

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

• A committee of 4 distinct roles (mayor, deputy mayor, secretary, treasurer) is chosen from 9 unique members.
• No member can hold more than one office, so repetition is not allowed.
• Since each office is distinct, the order of selection matters.

Determine the appropriate counting technique

Using the Permutations and Combinations knowledge points

• When repetition is not allowed and order matters, permutations must be used.
• Therefore, permutations apply because no item may be selected more than once and the order matters.

Answer:

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