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.

calculate how many different committees are possible.

(type a whole number.)

Explanation:

Analyze the selection constraints

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

• A city council has 9 distinct members.
• A 4-person executive committee is elected with specific roles: mayor, deputy mayor, secretary, and treasurer.
• Since each person can hold at most one office, repetition is not allowed (no item may be selected more than once).
• Since the roles are distinct, assigning the same 4 people to different roles results in a different committee; thus, order matters.

Determine the appropriate counting technique

Using the Permutations and Combinations knowledge points

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

Calculate the number of possible committees

Using the Permutations knowledge point

$$ P(9, 4) = \frac{9!}{(9-4)!} = \frac{9!}{5!} = 9 \times 8 \times 7 \times 6 = 3024 $$

Answer:

Question 1

• (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.

Question 2

The number of different committees is <blank>3024</blank>.