How many four character passwords can be created using the characters A, B, C, D, 1, and 2, using each once, where the first two characters are letters?
I tried the following.......
1st position can have 4 possibilities (any of A,B,C,D),
2nd position can have 3 possibilities (any 3 of A,B,C,D, since one is used),
3rd position can have 4 possibilities(any 2 of A,B,C,D or 1,2), and
4th position can have 3 possibilities.(any one of A,B,C,D or 1,2)
so $4 \times 3 \times 4 \times 3 = 144$ passwords can be created. The answer given is $288$. How do they get that?
What you have done looks right to me. Could it be a mistake in the question? It says "using each once", which doesn't make sense if it is a 4-character password. You've assumed they meant "using each at most once", but it could be that it should have said "how many six-character passwords"; the answer to that would be $288$ (your argument for the first four characters, then there are $2$ choices for the fifth and $1$ for the sixth).