Question

In the United States and Canada, a telephone number is a 10 -digit number of the form \(N X X-N X X-X X X X\) where \(N \in \mid 2,3, \ldots, 9\\}\) and \(X \in\\{0,1,2, \ldots, 9\\} .\) How many telephone numbers are possible? The first three digits of a telephone number are called an area code. How many different area codes must a city with 23,000,000 phones have? A previous scheme for forming telephone numbers required a format of \(N Y X \cdot N X X\) \(X X X X\) where \(N\) and \(X\) are defined as above and \(Y\) is either a 0 or a 1 . How many more phone numbers are possible under the new format than under the old format?

Short answer

640 billion numbers are possible; 3 area codes are needed; 51.2 billion more numbers possible now.

Step-by-step solution

Understanding the Divisions of a Telephone Number

A United States or Canadian telephone number is structured as \(NXX-NXX-XXXX\), where \(N\) is any digit between 2 and 9 for the area code and first digit of the exchange code, and \(X\) is any digit from 0 to 9. This gives us an initial perspective on the constraints for number formation.

Calculating the Total Possible Numbers in the Current Scheme

The total number of possible telephone numbers is calculated by multiplying the possibilities for each digit. The area code \(NXX\) consists of 8 choices for \(N\) (2 to 9) and 10 choices each for the two \(X\) digits, resulting in \(8 \times 10 \times 10 = 800\). The exchange code \(NXX\) follows the same logic, also resulting in 800. The last four \(X\) digits provide \(10^4 = 10000\) possibilities. Therefore, the total is \(800 \times 800 \times 10000 = 64,000,000,000\) possible numbers.

Determining Required Area Codes for 23,000,000 Phones

The number of area codes needed for 23,000,000 phones requires dividing the total phones by the number of phones an area code can accommodate, \(800 \times 10000 = 8,000,000\) (since last seven digits are unique within an area). Thus, \(\lceil 23,000,000/8,000,000 \rceil = 3\) area codes are required—rounded up to cover all phones.

Calculating Total Possible Numbers under Previous Scheme

In the previous scheme, telephone numbers follow the format \(NYX-NXX-XXXX\) where \(N\) and \(X\) are defined as before. The middle digit \(Y\) is a binary choice (0 or 1), providing 2 choices for \(Y\), \(8\) possibilities for the first \(N\), and 10 each for other \(X\) digits. Thus, the numbers are \(8 \times 2 \times 10 \times 800 \times 10000 = 12,800,000,000\).

Comparing the New and Old Formats

Subtract the previous scheme's total from the current scheme's total: \(64,000,000,000 - 12,800,000,000 = 51,200,000,000\). Therefore, 51.2 billion more numbers are possible with the current format.

Key concepts

Discrete Mathematics

Discrete mathematics is a branch of mathematics dealing with structures that are fundamentally discrete rather than continuous. This means it focuses on objects that can be counted, which include integers, graphs, and statements in logic, among other things. Unlike calculus or algebra, discrete mathematics does not involve continuous functions or variables. Instead, it uses a framework of distinct values.
Discrete mathematics plays a crucial role in computer science and telecommunications, providing tools for counting, graph theory, and logic. In the context of telephone number combinations, discrete mathematics helps us determine the number of possible arrangements and configurations given certain rules. By analyzing how numbers and symbols can be systematically organized, combinatorics, a key area within discrete mathematics, allows us to calculate the vast number of possible telephone numbers.

Telephone Number Combinations

Telephone number combinations involve the calculation of possible permutations of a set of digits conforming to specific rules. In North America, a telephone number is 10 digits long and follows the pattern of an area code, exchange code, and line number. The sequence is represented as \(NXX-NXX-XXXX\):

• \(N\) is a digit from 2 to 9, used for the area and exchange codes.
• \(X\) represents any digit from 0 to 9.

The challenge lies in combining digits in ways that meet these criteria. For each part of the phone number:

• The area code \(NXX\) has 800 options (given 8 choices for \(N\) and 100 choices for the \(XX\)).
• The exchange code \(NXX\) also yields 800 options.
• The line number \(XXXX\) has 10,000 possible combinations.

Multiplying these together gives a potential 64 billion phone numbers under the new system.

Area Code Calculation

An area code is a segment of a telephone number that designates a specific geographical region. Given a large number of phones in a city, like 23 million, one must calculate how many area codes are necessary to accommodate them all.
Each area code can provide service to 8 million phones because the sequence \(NXX-XXXX\) (800 options for \(NXX\), times 10,000 options for \(XXXX\)) can cover 8 million unique numbers. With 23 million phones, we perform the following division to find the necessary area codes:

• Divide 23,000,000 phones by 8,000,000 (the capacity of one area code).
• This results in 2.875. We round up to the nearest whole number because you cannot have a fraction of an area code.

Hence, 3 area codes are required to sufficiently cover 23 million phones, ensuring enough unique numbers are available.