Math CentralQuandaries & Queries


Question from Clayton:

Okay, so in this situation a license plate must have four letters (A-Z) and four numbers (0-9), ex. ABCD-1234, where repeating a letter or number is allowed ex. AAAA-1234, or ABCD-1111, or AAAA-1111. The order of letters first numbers second, or number first, letters second is allowable, and each state has its own plates, so ABCD-1234 from New York, and ABCD-1234 from Minnesota are considered different combinations, how many different license plates could there be?

Hi Clayton,

Think about generating a license plate with letters first followed by digits. You have 26 choices for the first character. Since letters can be repeated, regardless of your choice for the first letter you have 26 choices for the second letter. Thus you have 26 × 26 = 262 choices for the first two letters. Regardless of your choices so far you have 26 choices for the third letter and then again 26 choices for the fourth letter. Hence you have 264 choices for the 4 letter string of characters.

Now move on to the digits. You haven't said if zero is an allowed digit so I am not sure if you have 9 or 10 choices, but the procedure is the same as for the digits. Once you have calculated the number of possible license plates with letters followed by digits you need to double this as each possible plate can be written with the digits first and the letters next.

Finally you need to multiply by the number of states as each state has its own plates. I am not sure what the multiple is in this case. Does Washington DC get its own plates? What about Puerto Rico?


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