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.

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?

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

Explanation:

Identify the appropriate counting technique

Using the Arrangements with Repetitions knowledge point

• Selecting \(r\) items from a group of \(n\) choices where choices can be repeated and order matters.
• This corresponds to option A.

Calculate the number of possibilities for each part

Using the Fundamental Counting Principle knowledge point

• Area code \(aaa\): 3 digits, each cannot be 0 (9 choices per digit).

$$ 9 \times 9 \times 9 = 9^3 = 729 $$

• Prefix \(bbb\): 3 digits, each cannot be 9 (9 choices per digit).

$$ 9 \times 9 \times 9 = 9^3 = 729 $$

• Line number \(cccc\): 4 digits, no restrictions (10 choices per digit).

$$ 10 \times 10 \times 10 \times 10 = 10^4 = 10,000 $$

Compute the total number of telephone numbers

Using the Fundamental Counting Principle knowledge point

• Multiply the possibilities for each independent part:

$$ 729 \times 729 \times 10,000 = 531,441 \times 10,000 = 5,314,410,000 $$

• This corresponds to option A.

Answer:

Question 1

• (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.

Question 2

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