My book gives
$$20*48*47*46*45 $$
as the solution for the first part. I don't quite understand it. I am assuming $48*47*46*45 $ refers to the four other distinct cards that are not aces, but why do we multiply by 20?
The second part is given as
$$52*51*50*49*48—48*47*46*45*44$$
If I understand this part correctly, we are subtracting 5 non-aces from 5 distinct cards.
For the first part, choose a spot and choose a suit. There are $4\cdot 5=20$ ways of doing this. Then just choose non-aces like you said.
For the second part, it's actually choose 5 non-ace cards and subtract that from the total number of ways to choose 5 cards.