For years, telephone area codes in the United States and...

For years, telephone area codes in the United States and Canada consisted of a sequence of three digits. The first digit was an integer between 2 and $9,$ the second digit was either 0 or $1,$ and the third digit was any integer from 1 to $9 .$ How many area codes were possible? How many area codes starting with a 4 were possible?

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Key Concepts

Multiplication Principle

The multiplication principle is a fundamental concept in combinatorics stating that if an event can occur in a number of independent steps, the total number of outcomes is the product of the number of ways each individual step can occur. In this context, each digit of the area code represents an independent step with its own set of possibilities.

Counting with Restrictions

Counting with restrictions involves calculating the number of outcomes when specific conditions limit the choices available at each step. Here, each digit of the telephone area code has its own constraint, such as specific ranges or allowed values, and these restrictions must be incorporated into the multiplication calculation.

Conditional Counting

Conditional counting refers to adjusting the count of possibilities when additional conditions are applied to one or more of the steps. For example, when a digit is fixed to a particular value, the remaining digits are counted under their original restrictions, thereby reducing the total number of outcomes accordingly.

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