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, but the last two digits cannot both be 1 (to avoid confusion with area codes such as 911). 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.)

Verified step by step guidance

1. Step 1: Calculate the number of possible area codes. The first digit of the area code can be any digit from 2 to 9, giving 8 choices. The second and third digits can each be any digit from 0 to 9, giving 10 choices each. However, the last two digits cannot both be 1, so we subtract 1 from the total combinations of the second and third digits. The total number of area codes is therefore calculated as: (8 choices for the first digit) × (10 choices for the second digit) × (10 choices for the third digit) - 1 (to exclude the case where the last two digits are both 1).
2. Step 2: Calculate the number of possible exchange numbers. The same rules apply to the exchange numbers as to the area codes. The first digit can be any digit from 2 to 9 (8 choices), the second and third digits can each be any digit from 0 to 9 (10 choices each), and the last two digits cannot both be 1. The total number of exchange numbers is therefore calculated in the same way as the area codes: (8 choices for the first digit) × (10 choices for the second digit) × (10 choices for the third digit) - 1.
3. Step 3: Calculate the total number of 10-digit phone numbers. A 10-digit phone number consists of an area code, an exchange number, and four remaining digits. The four remaining digits can each be any digit from 0 to 9, giving 10 choices for each digit. The total number of 10-digit phone numbers is therefore: (number of possible area codes) × (number of possible exchange numbers) × (10 choices for the first remaining digit) × (10 choices for the second remaining digit) × (10 choices for the third remaining digit) × (10 choices for the fourth remaining digit).
4. Step 4: Compare the total number of possible phone numbers to the population. Given that the combined population of the United States, Canada, and a few islands is approximately 400,000,000, compare the total number of possible phone numbers to this population to determine if there are enough phone numbers available. Specifically, check if the total number of phone numbers exceeds the population.
5. Step 5: Conclude whether there are enough phone numbers. Based on the comparison in Step 4, determine if the total number of possible phone numbers is sufficient to meet the needs of the population. If the total number of phone numbers is significantly larger than the population, then there are enough phone numbers available.

Key Concepts

Combinatorial Counting

Combinatorial counting involves determining the number of ways to arrange or select items from a set, often using principles such as the multiplication rule. In this context, it helps calculate the total number of valid area codes and phone numbers by considering the restrictions on digit placement.

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics that deal with the arrangement and selection of items. For area codes and phone numbers, understanding how to apply these concepts allows for the calculation of different valid combinations of digits while adhering to the specified rules.

Population vs. Phone Number Capacity

This concept compares the total number of possible phone numbers to the population size to assess if there are enough unique phone numbers for everyone. By calculating the total number of valid 10-digit phone numbers, one can determine if the available combinations exceed the population of 400 million.