Trouble with permutations

passposs

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Dec 8, 2013
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I understand regular permutations, but let's say that I am trying to figure out the number of possible passwords, given that only letters can be used, and 10 available character spaces.

Its simple if these passwords -had- to be 10 characters long. But what about possible passwords if you can use less spaces than that too?

I'll use smaller numbers for an example:

Passwords can only consist of the characters A and B. The password can be up to 3 characters long. The usual 23 formula gives us 8 possibilities: AAA, AAB, ABA, ABB, BAA, BAB, BBA, BBB.

But I want to include AA, AB, BA, BB, A, and B in my answer, which would give me 14 possibilities. What would the formula for that be?
 
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