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

Explanation:

Analyze the selection conditions

Using the Permutations and Combinations knowledge points

• A city council has 7 members.
• We must elect a 4-person executive committee.
• The committee consists of specific, distinct roles: mayor, deputy mayor, secretary, and treasurer.
• Since each role is unique, assigning the same 4 people to different roles results in a different committee. Therefore, the order of selection matters.
• A single member cannot hold more than one office, meaning repetition is not allowed.

Determine the counting technique

Using the Permutations and Combinations knowledge points

• Since order matters and repetition is not allowed, permutations must be used.
• This matches option B: "Permutations should be used because no item may be selected more than once and the order matters."

Calculate the number of possibilities

Using the Permutations knowledge point

$$ P(7, 4) = \frac{7!}{(7-4)!} = \frac{7!}{3!} = 7 \times 6 \times 5 \times 4 = 840 $$

Answer:

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