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Express all probabilities as fractions. 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. 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 \(400,000,000 .\) )

Short Answer

Expert verified
There are 6,400,000,000 possible phone numbers, which is sufficient for the population of 400,000,000.

Step by step solution

01

Determine the number of possible area codes

The first digit of the area code can be any digit from 2 to 9, which gives us 8 options. The second and third digits can be any digit from 0 to 9, giving us 10 options each. Therefore, the number of possible area codes is calculated as: Number of area codes = 8 (options for the first digit) × 10 (options for the second digit) × 10 (options for the third digit) = 800
02

Determine the number of possible exchange numbers

Similar to the area code, the first digit of the exchange number can be any digit from 2 to 9 (8 options). The second and third digits can be any digit from 0 to 9 (10 options each). Therefore, the number of possible exchange numbers is: Number of exchange numbers = 8 (options for the first digit) × 10 (options for the second digit) × 10 (options for the third digit) = 800
03

Determine the number of possible line numbers

The last four digits can be any digit from 0 to 9. Since there are 10 options for each digit and there are 4 digits, the number of possible line numbers is calculated as: Number of line numbers = 10^4 = 10,000
04

Calculate the total number of possible 10-digit phone numbers

Combine the possible area codes, exchange numbers, and line numbers: Total phone numbers = (Number of area codes) × (Number of exchange numbers) × (Number of line numbers) Total phone numbers = 800 × 800 × 10,000 Total phone numbers = 6,400,000,000
05

Evaluate if the possible phone numbers are sufficient for the population

Given the population of 400,000,000 in the United States, Canada, and associated islands, the total number of possible phone numbers is 6,400,000,000. This is much larger than the population size, indicating that there are enough possible phone numbers. 6,400,000,000 (possible phone numbers) > 400,000,000 (population)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

area codes calculation
The area code is an essential part of a phone number, acting as an identifier for specific geographic regions. According to the rules provided, the area code consists of three digits.
The first digit can range from 2 to 9, providing 8 possible options. Each subsequent digit can be any number from 0 to 9, giving us 10 options each for the second and third digits.
Hence, if we multiply these options together, we calculate the total number of possible area codes:
\[\text{Number of area codes} = 8 \times 10 \times 10 = 800\]
This means that with the current rules, there are 800 possible area codes available for use.
exchange number calculation
Exchange numbers are the three digits that follow the area code in a phone number.
The rules for exchange numbers are similar to those for area codes.
The first digit can be any number between 2 and 9, providing 8 possible options. The second and third digits have 10 possible choices each, ranging from 0 to 9.
To find the total number of possible exchange numbers, we multiply the options together: \[\text{Number of exchange numbers} = 8 \times 10 \times 10 = 800\]
Thus, we have 800 unique combinations for the exchange numbers.
phone number probability
To determine how many possible 10-digit phone numbers exist, we need to consider the combinations of area codes, exchange numbers, and the last four digits of the phone number.
As previously calculated, there are 800 possible area codes and 800 possible exchange numbers.
For the last four digits, each digit can range from 0 to 9, giving us 10 options for each digit. Since there are four digits, we have: \[10^4 = 10,000\]
possible combinations for the last part of the phone number.

So, we combine all these possibilities: \[\text{Total phone numbers} = 800 \times 800 \times 10,000 = 6,400,000,000\]

Finally, we evaluate whether this number of phone numbers can accommodate the population of the United States, Canada, and surrounding islands, which is about 400,000,000. \[6,400,000,000 > 400,000,000\]
This indicates there are more than enough possible phone numbers to cover the population. Therefore, with the given rules, the probability of unique phone number allocation remains high.

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