Question

In Exercises 5–36, express all probabilities as fractions.

Phone Numbers Current rules for telephone area codes allow the use of digits 2–9 for the first digit, and 0–9 for the second and third digits. How many different area codes are possible with these rules? That same rule applies to the exchange numbers, which are the three digits immediately preceding the last four digits of a phone number. Given both of those rules, how many 10-digit phone numbers are possible? Given that these rules apply to the United States and Canada and a few islands, are there enough possible phone numbers? (Assume that the combined population is about 400,000,000.)

Short answer

The number of different area codes possible using the given rules is equal to 800.

The number of 10-digit phone numbers possible using the given rules is equal to 6,400,000,000.

Yes, there are enough possible phone numbers for the combined population of 400,000,000 people of the US and Canada.

Step-by-step solution

Given information

Different rules are applied to form 3-digit area codes and 10-digit phone numbers.

Counting principle

The number of different ways an event can occur is counted and multiplied using the given condition.

Here, two rules are provided for writing three digits:

Rule 1: The first digit is to be chosen from 2-9.

Rule 2: The second and third digits are to be chosen from 0-9.

Calculation

Area code (using rule 1 and rule 2):

The number of digits to choose from for the first digit of the area code = 8.

The number of digits to choose from for the second digit of the area code = 10.

The number of digits to choose from for the third digit of the area code = 10.

The total number of possible ways to write the area code:

$8\times 10\times 10=800$

Therefore, the number of different area codes possible is equal to 800.

Phone number:

As the same two rules apply to the three digits of the phone number preceding the last four digits, the total number of ways to write those three digits is equal to 800.

The number of ways to write the first three digits = 800.

For the remaining four digits, the total number of digits to choose from (0-9) = 10.

The number of ways to write the remaining four digits:

$10\times 10\times 10\times 10=10000$

Thus, the total number of possible phone numbers is:

$800\times 800\times 10000=6400000000$

Therefore, the number of different phone numbers possible is equal to 6,400,000,000.

Since the combined population of the US and Canada has 400,000,000 people, while the possible number of phone numbers is greater,there are sufficient phone numbers available for both countries.