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.

calculate how many different committees are possible.

(type a whole number.)

Explanation:

Identify the appropriate counting technique

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

• We must select 4 distinct members from a group of 7. No member can hold more than one office, so repetition is not allowed.
• Each selected member is assigned a specific, distinct role (mayor, deputy mayor, secretary, treasurer). Therefore, the order of selection matters.
• Since repetition is not allowed and order matters, permutations must be used.
• This corresponds to option B.

Calculate the number of possible committees

Using the Permutations knowledge point

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

Answer:

Question 1

• (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.

Question 2

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