Q:

How can you choose 4 distinct items from 5 things and get a repetition?

I am very confused about this question. 
You have a set of 5 letters {a,b,c,d,e} and you need the number of four letter strings that do not contain aa in the middle. 
This is 5 choose 4, but if you are choosing less items than the number that exist and all are distinct how can you possibly end up with a "aa" anywhere?

A:

Unrestricted, you have $5$ choices for each place in the string, hence $5^4$ strings.  
By "middle" I take it to be the central two, so strings like $-aa-$ are banned.
The other $2$ places could be filled in $5^2$ ways,
hence valid number of strings $= 5^4 - 5^2$ 

