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Question: License plates in British Columbia have the form ABC 123 or 123 ABC. All 26 letters (A to Z) and 10 digits (0 to 9) may be used more than once. How many plates are possible?

John Yu
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  • The claim seems false. E.g. http://www.bcpl8s.ca/images/Passenger/2014-2023/2017-EF084D(XL).jpg –  May 22 '18 at 16:10
  • split into two subproblems - for ABC 123; how many choices are there for each of the three places? what about for the three numbers? now look at 123 ABC and ask the same questions. multiply your two answers together. – thesmallprint May 22 '18 at 16:13
  • I highly discourage referring to things as "permutation questions" or "combination questions" because many times (after the first five examples) the solution will rely on neither! This is just a "multiplication principle question/rule of product question." It is more important to learn techniques than formulas. – JMoravitz May 22 '18 at 16:16
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    Pick whether the letter block appears first or whether the numeral block appears first (2 options). Pick what the leftmost letter in the letter block is (26 options). Pick what the middle letter in the letter block is (26 options). The final letter in the letter block (26 options). Pick the leftmost number in the number block (10 options) etc... multiply the number of options together – JMoravitz May 22 '18 at 16:21

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$2$ ways for the reversal of numbers and letters times $26^3$ ways for the letters times $10^3$ ways for the numbers.

$$2\cdot26^3\cdot10^3 = 35152000$$

Phil H
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  • But what about cases when there are letters or numbers that are the same. For example AAA111. Do I have to take into consideration distinguishable permutations? – John Yu May 22 '18 at 16:41
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    @John Yu The $26^3$ allows for letters to be the same. $26$ letters in the first position and $26$ letters in the second position includes all double letters. AA, BB, CC etc and so on for the 3rd letter and the same for the 3 numbers. This isn't a permutation or combination situation, just a counting exercise. – Phil H May 22 '18 at 16:47