This problem comes from a fun little branch of mathematics called "combinatorics." Basically, they want to know how many different combinations you could make out of the letters of the alphabet, given that they're all lowercase and no repeats are allowed. So let's put six boxes on our piece of paper, one for each character in the password. Got that?
_ _ _ _ _ _
Now, how many possible choices are there for the first character? We know the letters are all lowercase, so it would just be 26, since that's how many letters are in the alphabet. So the first character has 26 choices. Now, move on to the second character. Still 26 letters in the alphabet, but WAIT! We've already used one in the first position, and we know we can't repeat it. So now there are only 25 letters to choose from. Continue in this way to find the number of possibilities for each position in the password. You should end up with:
_ _ _ _ _ _
26 25 24 23 22 21
So now comes the fun part. If we know how many possibilities there are for each position in the password, we simply multiply them all together to get the total number of possible combinations. (There is a proof of why this works, but it's long and complicated so I won't get into it here.) Do that, and you'll get:
26 x 25 x 24 x 23 x 22 x 21 = 165,765,600
So there are 165,765,600 different password combinations using the parameters specified. Hope that helps!
~Ellen
10/22/2013

Ellen S.