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Chapter 6: Problem 20

In recent years, the state of California issued license plates using a combination of one letter of the alphabet followed by three digits, followed by another three letters of the alphabet. How many different license plates can be issued using this configuration?

Short Answer

Expert verified
There are 175,760,000 different license plates that can be issued using the given configuration, calculated by applying the counting principle: \( 26 * 10 * 10 * 10 * 26 * 26 * 26 \).

Step by step solution

01

Count Choices for First Position

The first position can be filled by any of the 26 letters of the alphabet. Therefore, there are 26 choices for the first position.
02

Count Choices for Second, Third, and Fourth Positions

The second, third, and fourth positions can be filled with any of the 10 digits (0-9). Hence, there are 10 choices for each of these positions.
03

Count Choices for Fifth, Sixth, and Seventh Positions

The fifth, sixth, and seventh positions can be filled with any of the 26 letters of the alphabet. Therefore, there are 26 choices for each of these positions.
04

Apply Counting Principle

The counting principle states that we multiply the number of choices available per position to find the total number of combinations. Thus, the total number of different license plates will be: Total combinations = First_position_choices * Second_position_choices * Third_position_choices * Fourth_position_choices * Fifth_position_choices * Sixth_position_choices * Seventh_position_choices
05

Calculate Total Combinations

Substitute the values calculated earlier for the choices for each position: Total combinations = 26 * 10 * 10 * 10 * 26 * 26 * 26
06

Compute the Result

The final multiplication to obtain the total number of different license plates: Total combinations = 175,760,000 Therefore, there are 175,760,000 different license plates that can be issued using the given configuration.

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