Probability: Arrangements Question

Question: 5 people designated as A, B, C, D, E are arranged in linear order. Assuming that each possible order is equally likely, what is the probability that

a) there is exactly one person between A and B

b) there is exactly 2 people between A and B

c) there is exactly 3 people between A and B

I am not sure how to work out this problem. I am having trouble coming out with the possible ways of outcomes and I do not know what to do with them. Here is what I have so far:

a) there is exactly one person between A and B

There is 3 ways that will happen ACBDE, ADBCE, AEBCD

but will ACBED count as another way?

Then there is 120 ways that these 5 people can line up.

Would we have to divide 3 by 120 to get the proability?

Thank you very much for your help.