Question

answer the following question using the appropriate counting technique, which may be either arrangements with repetition, permutations, or combinations. be sure to explain why this counting technique applies to the problem.

how many different telephone numbers of the form aaa-bbb-cccc can be formed if the area code aaa cannot contain 0 and the prefix bbb cannot contain 9?

what counting technique should be used to make this calculation?

a. arrangements with repetitions because there are \\(r\\) selections from a group of \\(n\\) choices and choices can be repeated.
b. combinations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement does not matter.
c. arrangements with repetitions because the selections come from a single group of items, and the order of the arrangement matters.
d. permutations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement matters.

Explanation:

Analyze the problem requirements

The question asks to identify the appropriate counting technique for forming telephone numbers of the form aaa-bbb-cccc.

Evaluate repetition and order

Each digit in a telephone number is selected from the set of digits 0–9. Digits can be repeated (e.g., the area code can be 555), and the order of the digits matters because changing the order results in a different telephone number.

Match with definitions

Using Combinations, order does not matter and items cannot be repeated. For permutations, items cannot be repeated. Since digits can be repeated and order matters, we must use arrangements with repetitions.

Select the correct option

Option A states: "Arrangements with repetitions because there are \(r\) selections from a group of \(n\) choices and choices can be repeated." This matches our analysis.

Answer:

• A. Arrangements with repetitions because there are r selections from a group of n choices and choices can be repeated. (Correct answer)
• B. Combinations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement does not matter.
• C. Arrangements with repetitions because the selections come from a single group of items, and the order of the arrangement matters.
• D. Permutations because the selections come from a single group of items, no item can be selected more than once and the order of the arrangement matters.