Problem 35 - Algebra and Trigonometry

Question

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 was 144.

Step 1: Calculate the possibilities for the first digit

The first digit can be any number from 2 through 9. Therefore, there are 8 possibilities in total, i.e. {2,3,4,5,6,7,8,9}.

Step 2: Calculate the possibilities for the second digit

The second digit can either be 0 or 1. So, there are 2 possibilities in total, i.e. {0,1}.

Step 3: Calculate the possibilities for the third digit

The third digit can be any number except 0. Hence, there are 9 possibilities in total, i.e. {1,2,3,4,5,6,7,8,9}.

Step 4: Apply the counting principle

We can now apply the counting principle to find the total number of area codes. This principle states that if one event can occur in m ways and a second can occur independently in n ways, then the two events can occur together in m*n ways. So the total number of different area codes = number of possibilities for first digit * number of possibilities for second digit * number of possibilities for third digit.

Step 5: Calculate the total number of area codes

Substitute the values from steps 1 to 3 into the formula from step 4 to get the total number of area codes. This gives a total of 8 possibilities for the first digit * 2 possibilities for the second digit * 9 possibilities for the third digit = 144 different area codes.

Key Concepts

Counting Principle

The counting principle is a fundamental concept in combinatorics. It allows us to determine the total number of outcomes for a sequence of events by multiplying the number of choices available for each event. For instance, if you have a choice between two shirts and three pairs of pants, you can pair each shirt with each pair of pants, giving you a total of 6 unique outfit combinations.
This principle is especially useful when dealing with problems that involve multiple stages, where each stage or selection is independent of the others. In the case of area codes, we are dealing with three separate choices: selecting the first digit, the second digit, and the third digit.

• The first digit can be any of the numbers from 2 to 9, offering 8 possible choices.
• The second digit, limited to either 0 or 1, provides 2 possibilities.
• Finally, the third digit can be any number except 0, giving 9 possible options.

By applying the counting principle, we multiply these possibilities together: 8 (first digit) * 2 (second digit) * 9 (third digit) to find the total number of area codes possible.

Digit Possibilities

Digit possibilities refer to the number of different values a particular digit can have in a sequence, typically in numerical codes or combinations. It's essentially exploring how many choices are available for each digit position.
Consider the concept of setting rules for each digit as seen in the exercise. In the 1945 area code plan:

• The first digit was restricted to numbers ranging from 2 to 9, translating to 8 total possibilities.
• The second digit was restricted to either 0 or 1, providing 2 possibilities.
• The third digit could be any number except 0, offering 9 choices.

Understanding digit possibilities helps in structuring problems where certain conditions or restrictions limit the options for each digit, shaping how you apply the counting principle effectively to find total combinations.

Area Codes Calculation

Calculating different area codes is an example of applying counting principles in real-world scenarios. In historical context, area codes are numerical codes assigned to specific geographic regions to facilitate long-distance calling. Back in 1945, they were structured using specific rules to ensure uniqueness and avoid confusion.
Let's examine the calculation process:

• First, identify the rule for each digit position in the area code. In this setup, only digits 2 through 9 could be used first, allowing for 8 choices.
• The second digit was limited to the two options: 0 or 1.
• Finally, the third digit was open to 1 through 9, presenting 9 possible numbers.

The total number of area codes is then calculated by multiplying the number of possibilities for each digit position. Thus, 8 * 2 * 9 results in 144 unique area codes. This methodical approach illustrates the applications of combinatorics in planning and structuring systems efficiently.