First, suppose we had 20 seconds and needed to guess. You can get down to two answers quite quickly. We're making a seating arrangement with 5 people. If there are no restrictions at all, the answer would be 5! = 120. There are restrictions, so the answer must be less. If we ignore the restriction about the daughters, and only note that a parent must be in the driver's seat, we would have 2 choices for that seat, and 4! choices for the rest. That is, we'd have 2*4! = 48 arrangements. But we have the restriction about the daughters, so the answer must be less than 48. That gets us to A) and B) only.
If you want to do this question completely, I don't see any 25-second solution. I think you do need to look at cases. I'll write P, D and S for parent, daughter and son:
1. We could have:
Front: PD
Back: P, D, S
Then we have 2 choices for the driver, and 2 for the other front seat = 4 choices in total (multiply your number of choices). We also have 3*2*1 choices for the back seat, since they can sit in any order. Thus, we have 2*2*3*2*1 = 24 arrangements.
2. We could have:
Front: PP
Back: D, D, S
We have only 2 choices for the front seat (either parent could drive). In the back, the son needs to be in the middle, so there are only 2 arrangements in the back: the daughters could be on either side. There are 2*2 = 4 arrangements.
3. Finally we could have:
Front: PS
Back: D, P, D
Again, we have two choices for the driver. We also have only 2 choices for the back row, because the remaining parent must be in the middle. 2*2 = 4 arrangements.
24+4+4 = 32.
