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

Question

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?

Answer 1

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 $= 5^4 - 5^2$

Answer 2

There are $5^2-1$ options for the two letters in the middle and then $5^2$ options for the letters in the edges. So there are $(5^2-1)\times5^2=24\times 25=600$ words.