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I am trying to understand the product rule and I have a simple example it says,

 If I have a license plate with two English letters how many different plates
 can be made?

The answer is 26^2

Now another question is in the same format but it asks how many plates can be made with upper and lower case letters. Would this still be counted as 26^2? Or would this be 52^2? Or are they all different sets, making it 78^2? I looked through lecture slides and they were no help. Thanks again.

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That is not a permutation. You should google "Variation with repetition" – SJuan76 Jun 16 '13 at 16:49
up vote 0 down vote accepted

To think about the problem intuitively, imagine just choosing one letter out of the set of upper and lower case letters. You have $26$ lower case letters and $26$ upper case letters, giving you $26+26=52$ characters in total. Therefore there are $52$ potential choices for the first character.

Choosing for the second letter is exactly the same, and so you can choose any of the $52$ characters; combining these is a simple matter of multiplication (because for each of the $52$ choices for the first character there are $52$ choices for the second character). Therefore we have:

$$52\times52=52^{2}=2704\text{ combinations}$$

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Thanks, my instructor for summer is teaching his first class and he doesn't stretch his time out well, leaving him to rush over 50 slides in 15 minutes! – Hermes Trismegistus Jun 16 '13 at 16:58
@HermesTrismegistus No problem, glad to help! If this answered your question could you click on the tick next to the answer to mark the question as answered, so it no longer shows up in the unanswered section? Thanks! – Shaktal Jun 16 '13 at 17:04

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