WarningMy responses to combinations questions are notoriously inaccurate

How many ways can 12 prizes be awarded amongst 4 people.

12!/3!

You know that each person gets 3 prizes.

There are 12! ways of distributing the prizes, and each person gets 3 of them. But the order of the 3 prizes receieved doesn't matter. there are 3! ways of ordering the set of 3 prizes receieved

From a deck of 52, how many 4 card hands include 1 from each suit

There are 4 suits in a standard deck of cards. So you must choose 1 of the 13 spades, then 1 of the 13 clubs,....etc

There are 13*13*13*13 ways of doing that. However the order does not matter so the final answer is

13*13*13*13 / 4!

(there are 4! ways of ordering the 4 suits)