Question

Is there a fast way to list all the possible permutations for the letters A B C D? This also includes AAAA, BBBB, CCCC, DDDD, ABBB, ACCC, ADDD, etc

Answer 1

permutations does not normally include identical elements
abcd is ok
as is abdc
but using a twice is not part of it
there is no fast way to do what you asked that I know of

Answer 2

This link will not help with your listing task but does show how to do the calculation with repeating elements.
http://www.mathwarehouse.com/probability/permutations-repeated-items.php

Answer 3

If you want permutations of elements, that implies no duplications, unless there are duplicate elements.

If you just wall all possible 4-letter strings of ABCD, then there are 4 choices for each position in the string: 4^4 = 256

If you want all strings of length up to 4, then there are

4 + 4^2 + 4^3 + 4^4 = 4(4^4-1)/(4-1) = 340