This kind of exercises are solved by counting the choices you have at every step.
We have five possible letters: A,B,C,D and E. We want to form a two-letter password without repetitions. This means that we have five choices for the first letter: in can be any of the five letters, because we have no restrictions so far.
Now, assume we have chosen a particular letter to be the first one. How many choices do we have for the second letter? Well, we're told that we can't repeat letters, so we have four choices. In fact, once the first letter is fixed, the second letter can be any of the five, except the one we fixed as first.
For example, if the first letter is C, we have the following four choices: CA, CB, CD, CE, since we're not allowed to choose CC, because there are repeating letters.
So, we have four choice for the second letter, for each of the five choices we have for the first letter. This leads to a total of 20 passwords:
First letter A: AB, AC, AD, AE
First letter B: BA, BC, BD, BE
First letter C: CA, CB, CD, CE
First letter D: DA, DB, DC, DE
First letter E: EA, EB, EC, ED
