Problem 35 - College Algebra

Question

Use the Fundamental Counting Principle to solve In the original plan for area codes in \(1945,\) the first digit could be any number from 2 through \(9,\) the second digit was either 0 or \(1,\) and the third digit could be any number except \(0 .\) With this plan, how many different area codes were possible?

Step-by-Step Solution

Answer

According to the Fundamental Counting Principle, there were 144 possible area codes.

Step 1: Determine the Number of Ways for the First Digit

The first digit of the area code can be any number from 2 through 9. This provides 8 different possibilities (2, 3, 4, 5, 6, 7, 8, 9).

Step 2: Determine the Number of Ways for the Second Digit

For the second digit of the area code, it can either be 0 or 1. This narrows it down to 2 possibilities.

Step 3: Determine the Number of Ways for the Third Digit

The third digit of the area code can be any number except 0. Hence, there are 9 different possibilities (1, 2, 3, 4, 5, 6, 7, 8, 9).

Step 4: Applying the Fundamental Counting Principle

Now, by the Fundamental Counting Principle if there are p ways to do one thing, q ways to do another, and r ways to do a third, there are pqr ways to do all three. Thus, there are 8*2*9 = 144 different possible area codes.

Key Concepts

area codes

Area codes are essential elements of telephone number systems, especially in large countries, as they help in routing calls to specific regions. This system was primarily introduced to manage the vast number of phone numbers needed as the popularity of telephones grew, primarily in the mid-20th century. Area codes typically come before the main telephone number and are crucial for identifying the geographical location of the dialed number. In the United States, for example, the North American Numbering Plan (NANP) employs a systematic approach to area codes.
In 1945, when the initial plan for area codes was introduced, they had to follow specific rules for digit combinations. The structure aimed to ensure clarity and prevent confusion in dialing. The first digit ranged from 2 to 9, the second was either 0 or 1, and the third could not be 0. These constraints allowed efficient communication and reduced misdialing when numbers were manually dialed.

probability

Probability is a branch of mathematics concerned with the likelihood of events occurring. It is incredibly useful in predicting outcomes and making informed decisions. When discussing probability, the fundamental aspect is understanding all the possible outcomes and how likely each is to occur.
In the context of area codes, the probability framework helps determine the potential combinations by leveraging the Fundamental Counting Principle. However, it is important to note that this specific exercise isn't directly about probability since we aren't weighing how likely each area code is to be assigned; rather, we're identifying all the possible combinations available under certain conditions.

combinatorics

Combinatorics is a fascinating field of mathematics that deals with counting, arrangement, and combination of elements within sets. It plays a vital role in a variety of disciplines, from solving theoretical puzzles to optimizing computing processes.
When considering the problem of calculating how many different area codes can exist given specific rules, combinatorics is at the heart. The fundamental counting principle is a combinatorial method that aids in counting effectively without having to list every possible arrangement explicitly. By determining the number of possibilities for each digit of an area code and then multiplying them, combinatorics helps find the total number of distinct configurations.

number systems

Number systems are critical frameworks in mathematics, allowing us to flexibly and efficiently represent values. In daily life, we encounter several kinds of number systems including the decimal system, binary system, and others.
The original exercise is closely related to number systems since it involves the use of specific digits (2-9, 0, 1) to form area codes. These digits are essentially part of the decimal system, which is the most commonly used. Understanding the characteristics of number systems, such as base and range, is essential when constructing and interpreting sequences, like area codes here, and plays an important role in telecommunications and digital processing.