Question

Phone numbers in North America have 10 digits: a threedigit area code, a three-digit exchange number, and the four final digits that make each phone number unique. Neither area codes nor exchange numbers can start with 0 or $1 .$ Prior to 1994 the second digit of the area code had to be a 0 or 1 Sixteen area codes are reserved for special services (such as 911 and 411 ). In $1994,$ the last area code was used up and the rules were changed to allow the digits 2 through 9 as the middle digit in area codes. How many different phone numbers were possible prior to $1994 ?$

Name: Problem 86, Chapter 10: Precalculus
Uploaded: 2020-05-24T16:40:34-08:00
Duration: 33 s
Channel: Elizabeth Xu

   Phone numbers in North America have 10 digits: a threedigit area code, a three-digit exchange number, and the four final digits that make each phone number unique. Neither area codes nor exchange numbers can start with 0 or $1 .$ Prior to 1994 the second digit of the area code had to be a 0 or 1 Sixteen area codes are reserved for special services (such as 911 and 411 ). In $1994,$ the last area code was used up and the rules were changed to allow the digits 2 through 9 as the middle digit in area codes.
How many different phone numbers were possible prior to $1994 ?$

Precalculus

John W. Coburn 2nd Edition

Chapter 10, Problem 86

Instant Answer

Solved by Expert Elizabeth Xu

05/24/2020

Step 1

Therefore, we have 8 options for the first digit. Prior to 1994, the second digit of the area code had to be a 0 or 1, so we have 2 options for the second digit. The third digit can be any number from 0 to 9, so we have 10 options for the third digit. However, 16 Show more…

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Phone numbers in North America have 10 digits: a threedigit area code, a three-digit exchange number, and the four final digits that make each phone number unique. Neither area codes nor exchange numbers can start with 0 or $1 .$ Prior to 1994 the second digit of the area code had to be a 0 or 1 Sixteen area codes are reserved for special services (such as 911 and 411 ). In $1994,$ the last area code was used up and the rules were changed to allow the digits 2 through 9 as the middle digit in area codes. How many different phone numbers were possible prior to $1994 ?$

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Key Concepts

Counting with Restrictions

Counting with restrictions involves determining the number of valid outcomes when certain positions or choices are limited by additional rules. In combinatorial problems, specific digits or elements may be excluded from being selected for certain positions, which reduces the total number of outcomes. Understanding how to apply these restrictions correctly is key to solving problems where not every combination is allowed.

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The multiplication principle is a fundamental counting rule stating that if one event can occur in m ways and a second independent event can occur in n ways, then the two events together can occur in m × n ways. This principle is essential in counting problems where a procedure is broken into independent sequential steps, and the total number of outcomes is obtained by multiplying the number of ways each step can occur.

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