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 five members must elect a three-person executive committee consisting of a mayor, deputy mayor, and treasurer. how many executive committees are possible?

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. arrangements with repetition should be used because we make selections from a group of choices.
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 no item may be selected more than once and the order matters.
e. permutations should be used because we make selections from a group of choices.

state how many different committees are possible.
(type a whole number.)

Explanation:

Analyze the selection conditions

Using the Permutations and Combinations knowledge points:

• A city council has 5 members.
• We must elect a 3-person executive committee consisting of specific, distinct roles: a mayor, a deputy mayor, and a treasurer.
• Since each person can hold at most one office, no member can be selected more than once (no repetition).
• Because the roles (mayor, deputy mayor, treasurer) are distinct, the order in which we assign the members to these roles matters.

Determine the counting technique

Using the Permutations and Combinations knowledge points:

• When order matters and repetition is not allowed, we use permutations.
• Therefore, 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:

• We need to find the number of permutations of 5 distinct items taken 3 at a time:

$$ P(5, 3) = \frac{5!}{(5-3)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 5 \times 4 \times 3 = 60 $$

Answer:

Question 1

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

Question 2

60