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.

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

Explanation:

Identify the counting technique

Using the Arrangements with Repetitions knowledge point

• Telephone numbers consist of digits chosen from the set \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\).
• Digits can be repeated within the number, and the order of the digits matters.
• Therefore, we make \(r\) selections from a group of \(n\) choices where choices can be repeated.

Calculate the number of possibilities

Using the Fundamental Counting Principle knowledge point

• The telephone number has the form \(aaa-bbb-cccc\), which consists of 10 digits in total.
• For the area code \(aaa\) (3 digits): since it cannot contain 0, each of the 3 positions has 9 possible choices \(\{1, 2, 3, 4, 5, 6, 7, 8, 9\}\).

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

• For the prefix \(bbb\) (3 digits): since it cannot contain 1, each of the 3 positions has 9 possible choices \(\{0, 2, 3, 4, 5, 6, 7, 8, 9\}\).

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

• For the line number \(cccc\) (4 digits): there are no restrictions, so each of the 4 positions has 10 possible choices \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\).

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

• Total combinations:

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

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.