Problem 27 - College Algebra

Question

Use the Fundamental Counting Principle to solve Exercises \(21-32\). 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

The total number of different area codes possible is 144

Step 1: Identifying Options for the First Digit

The first digit, X, can be any number from 2 through 9. This gives us 8 options in total.

Step 2: Identifying Options for the Second Digit

The second digit, Y, can be either 0 or 1, so we have 2 options.

Step 3: Identifying Options for the Third Digit

The third digit, Z, can be any number from 1 through 9 (since 0 is excluded). So, there are 9 options for the third digit.

Step 4: Applying the Fundamental Counting Principle

Now, we multiply the options for every digit to find out the total number of different area codes possible. Hence, the total count of possible area codes is \( 8 \times 2 \times 9 \)

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics focused on counting, arranging, and selecting objects. It is the mathematical study of finite or countable structures. In essence, it's all about figuring out how many ways we can combine certain items. This plays a crucial role in solving problems where we need to count the number of possible outcomes or arrangements. Combinatorics gives us the tools to handle questions related to permutations (where order matters) and combinations (where order doesn't matter).

In the context of our area code problem, we use combinatorics to determine the total number of possible area codes. By recognizing each position in the area code as an independent choice, we apply combinatorics principles to count all possible combinations. The Fundamental Counting Principle (FCP), a fundamental concept in combinatorics, enables us to efficiently compute the total number of outcomes from multiple choices or events.

Area Codes

Area codes are essential components of phone numbers in many regions. They help to identify specific geographical areas within a country. Initially, area codes were simple and designed with certain restrictions. For example, in the original 1945 plan, area codes consisted of three digits with specific rules:

• The first digit could be any number from 2 through 9.
• The second digit was limited to 0 or 1.
• The third digit could be any number from 1 through 9, but not 0.

These rules were established to reduce confusion and maximize the efficiency of the dialing system. By adhering to these constraints, different regions could be clearly demarcated, allowing for greater national and international communication without overlapping area designations. Understanding these historical limitations helps us appreciate how earlier telecommunications dealt with geographical numbering and the role of clear combinatorial applications in real-life solutions.

Mathematical Counting Methods

Mathematical counting methods, like the Fundamental Counting Principle, are indispensable when solving problems that require tallying possible outcomes. The Fundamental Counting Principle states that if we have several events, where each event has a particular number of ways it can occur, the total number of ways all events can happen is the product of the number of ways each individual event can occur.

Let's break down the principle with our area code exercise:

• The first digit has 8 options (2-9).
• The second digit has 2 options (0 or 1).
• The third digit has 9 options (1-9).

According to the Fundamental Counting Principle, the total number of different area codes is calculated by multiplying the options for each digit: \[8 \times 2 \times 9 = 144.\]

Thus, there were 144 unique combinations available for area codes, thanks to the restrictions in the 1945 plan. By applying these counting methods, we can effortlessly uncover the vast number of possibilities in structured settings.