How many different 5-card hands can be dealt from a standard 52-card deck?
I've highlighted the word different because I believe it's the key to the question, am I right in thinking the answer is
No, they wouldn't. Your answer, 52*51*50*49*48, which is the same as or which includes the same cards in different orders. To not count the same cards in different orders as different hands, you need to divide by 5!, the number of orders of the same 5 cards. That will give you which is exactly as Plato said.