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 6 and the prefix bbb cannot contain 1?

what counting technique should be used to make this calculation?

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

how many different telephone numbers of the form aaa-bbb-cccc can be formed if the area code aaa cannot contain 6 and the prefix bbb cannot contain 1?

a. there are 6,561,000,000 different telephone numbers.
b. there are 4,782,969,000 different telephone numbers.
c. there are 5,314,410,000 different telephone numbers.
d. there are 387,420,489 different telephone numbers.

Explanation:

Identify the counting technique

To determine the correct counting technique, we analyze the structure of a telephone number of the form aaa-bbb-cccc. Each digit is selected from the set of digits \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Since digits can be repeated within the number (for example, the area code can be 555) and the order of the digits matters (555-123-4567 is different from 555-321-4567), we are making \(r\) selections from a group of \(n\) choices where choices can be repeated. This corresponds to Arrangements with Repetitions.

Calculate choices for each section

Using the Fundamental Counting Principle, we calculate the number of possibilities for each part of the telephone number aaa-bbb-cccc:

• Area code (aaa): This consists of 3 digits. Normally, each digit has 10 choices \(\{0-9\}\). However, the area code cannot contain the digit 6. Thus, each of the 3 digits has \(10 - 1 = 9\) choices.

$$ \text{Choices for aaa} = 9 \times 9 \times 9 = 9^3 = 729 $$

• Prefix (bbb): This consists of 3 digits. The prefix cannot contain the digit 1. Thus, each of the 3 digits has \(10 - 1 = 9\) choices.

$$ \text{Choices for bbb} = 9 \times 9 \times 9 = 9^3 = 729 $$

• Line number (cccc): This consists of 4 digits. There are no restrictions on these digits, so each has 10 choices.

$$ \text{Choices for cccc} = 10 \times 10 \times 10 \times 10 = 10^4 = 10,000 $$

Compute total telephone numbers

Using the Fundamental Counting Principle, we multiply the number of possibilities for each independent section to find the total number of unique telephone numbers:

$$ \text{Total numbers} = 729 \times 729 \times 10,000 $$

$$ 729 \times 729 = 531,441 $$

$$ \text{Total numbers} = 531,441 \times 10,000 = 5,314,410,000 $$

This matches option C of the second question.

Answer:

Question 1

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

Question 2

• A. There are 6,561,000,000 different telephone numbers.
• B. There are 4,782,969,000 different telephone numbers.
• C. There are 5,314,410,000 different telephone numbers. (Correct answer)
• D. There are 387,420,489 different telephone numbers.