How many different arrangements of A, B, C, D, and E are possible wher

This is a hard question.
We'll calculate it by calculating all the possible arrangements, subtracting the 'bad' arrangements and correcting for double-counting.
This is similar to calculating a difficult probability by calculating 1 minus the probability and is a Precise approach.

There are 5! = 120 total options to arrange the 5 letters.
The 'bad' arrangements are those that have an A and a B adjacent or a D and an E adjacent.
To calculate this we'll first 'stick' A and B together to form 'one letter'.
Then we have 4 new letters - 'AB', C, D, E.
There are 4! = 24 ways to arrange these letters
Similarly, each of the combinations of
'BA', C, D, E
A, B, C, 'DE'
A, B, C, 'ED'
has 24 options giving a total of 24*4 = 96 'bad' options.
So far we've counted 120 - 96 = 24 options.
But - we've doublecounted! We've removed arrangements with both A,B adjacent and D,E adjacent twice!
We'll add these in similarly to what we did above:
If 'AB' and 'DE' are both together then we have 3 'letters' we can rearrange:
'AB', C, 'DE' : 3! = 6 options.
'AB', C, 'ED' : 3! = 6 options.
'BA', C, 'DE' : 3! = 6 options.
'BA', C, 'ED' : 3! = 6 options.
So we need to add back 6*4 = 24 giving a final total of 24 + 24 = 48 options.
(B) is our answer.

** It is also possible to brute force list everything but with 5 letters and all the constraints it is probably more dangerous than useful.