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 1?

what counting technique should be used to make this calculation?

a. 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.
b. arrangements with repetitions because the selections come from a single group of items, and the order of the arrangement matters.
c. arrangements with repetitions because there are \\(r\\) selections from a group of \\(n\\) choices and choices can be repeated.
d. 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.

Explanation:

Analyze the characteristics of telephone numbers

To determine the correct counting technique, we must analyze how telephone numbers of the form aaa-bbb-cccc are constructed.

• Each digit in a telephone number is selected from the set of digits \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\).
• Digits can be repeated within the area code, prefix, or line number (for example, 555-555-5555 is a valid structure).
• The order of the digits matters because changing the order of the digits results in a completely different telephone number.

Identify the appropriate counting technique

Using the Arrangements with Repetitions concept:

• We are making \(r\) selections (the digits of the phone number) from a group of \(n\) choices (the available digits \(0\) through \(9\)).
• Since digits can be repeated and the order of arrangement matters, this scenario represents arrangements with repetitions.

Evaluate the given multiple-choice options

• Option A suggests permutations, which assumes no item can be selected more than once. This is incorrect because digits can be repeated.
• Option B suggests arrangements with repetitions because selections come from a single group of items and no item can be selected more than once. This is a contradiction since arrangements with repetitions allow repeating items.
• Option C states: "Arrangements with repetitions because there are \(r\) selections from a group of \(n\) choices and choices can be repeated." This correctly defines the technique and its justification.
• Option D suggests combinations, which assumes order does not matter and items cannot be repeated. This is incorrect.

Answer:

• A. 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.
• B. Arrangements with repetitions because the selections come from a single group of items, and the order of the arrangement matters.
• C. Arrangements with repetitions because there are r selections from a group of n choices and choices can be repeated. (Correct answer)
• D. 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.