License Plate Combinations: Calculating the Total Possible Plates in a State's System

License plates are a vital component in many state's vehicle registration systems. Understanding the total number of possible combinations is crucial for planning and management. In a certain state, license plates consist of a sequence of zero to three letters followed by a sequence of zero to four digits, with the provision that a blank plate is not allowed. Let's dive into the detailed breakdown of how this system works and calculate the total number of different license plates that can be produced.

Step 1: Counting the Letters

The first step in our calculation is to determine the number of possible letter combinations. The license plates can have from 0 to 3 letters, each letter from A to Z, which gives us 26 possible options for each letter. We can calculate the total combinations as follows:

0 letters: 1 way (the empty string) 1 letter: 26 ways (one of the 26 letters) 2 letters: 26^2 676 ways (any combination of two letters) 3 letters: 26^3 17576 ways (any combination of three letters)

By summing these values, we get the total number of possible letter combinations:

1 26 676 17576 18279 ways

Step 2: Counting the Digits

The second part of our calculation involves counting the possible digit combinations. The license plates can have from 0 to 4 digits, with each digit ranging from 0 to 9. We can calculate the total combinations as follows:

0 digits: 1 way (the empty string) 1 digit: 10 ways (one of the 10 digits: 0-9) 2 digits: 10^2 100 ways (any combination of two digits) 3 digits: 10^3 1000 ways (any combination of three digits) 4 digits: 10^4 10000 ways (any combination of four digits)

By summing these values, we get the total number of possible digit combinations:

1 10 100 1000 10000 11111 ways

Step 3: Total Combinations

Now that we have the total number of letter and digit combinations, we can calculate the total number of possible license plates. However, since a blank plate is not allowed, we need to subtract the one case where both are zero. Therefore, the formula for the total combinations is:

Total combinations 18279 ways (letters) x 11111 ways (digits) 203091369 ways

However, we must subtract the one case where both are zero (1 way) to exclude the blank plate.

Thus, the final total is:

203091369 - 1 203091368 ways

Conclusion

In conclusion, the total number of different license plates that the state can produce, considering the given format and excluding the blank plate, is:

203091368

This detailed breakdown helps in understanding the complexity of the system and the vast number of possible license plates. It is essential for state authorities to manage this system effectively to ensure efficient vehicle registration and identification.