Question

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Explanation:

Identify the appropriate counting technique

To determine the number of telephone numbers of the form aaa-bbb-cccc, we must choose digits from the set of integers \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Since digits can be repeated in a telephone number (for example, the area code can be 999) and the order of the digits matters, we use Arrangements with Repetitions. This technique applies because we make \( r \) selections from a group of \( n \) choices where choices can be repeated.

Calculate the number of choices for each part

Using the Fundamental Counting Principle, we calculate the number of possibilities for each section of the telephone number:

• Area code (aaa): This consists of 3 digits. Since it cannot contain 0, each digit has 9 possible choices \( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \).

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

• Prefix (bbb): This consists of 3 digits. Since it cannot contain 1, each digit has 9 possible choices \( \{0, 2, 3, 4, 5, 6, 7, 8, 9\} \).

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

• Line number (cccc): This consists of 4 digits with no restrictions, so each digit has 10 possible choices \( \{0, 1, \dots, 9\} \).

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

Compute the total number of telephone numbers

Multiply the possibilities for each independent section together:

$$ \text{Total} = 729 \times 729 \times 10,000 = 531,441 \times 10,000 = 5,314,410,000 $$

Answer:

Question 1

• (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.

Question 2

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