Let's take this one type of hand at a time.
Five of the same card: There are 52 different cards in a deck, for 52 combinations.
Four of the same card: There are 52 possible cards for the four that are the same and 51 for the singleton. So, there are 52*51 = 2,652 possible combinations for a four of a kind.
Three of the same card and two of another card: There are 52 possible cards for the three that are the same and 51 for the pair. So, there are 52*51 = 2,652 possible combinations for a full house.
Three of the same card and two different singletons : There are 52 possible cards for the three that are the same and combin(51,2)=1,275 for the two singletons. So, there are 52*1,275 = 66,300 possible combinations for a three of a kind.
Two pairs of the same card and one singleton : There are combin(52,2)=1,326 combinations for the two different cards that form pairs. Then there are 50 left for the singleton. So, there are 1,326*50=66,300 possible combinations for a two pair.
One pair of the same card and three singletons: There are 52 possible cards for the pair. Then there are combin(51,3)=20,825 ways to choose 3 cards out of the 51 left for the singletons. So, the number of combinations for a pair is 52*20,825 = 1,082,900.
Five singletons: There are combin(52,5)=2,598,960 ways to choose five cards out of 52.
The sum of all these combinations is 3,819,816.
This question is asked and discussed in my forum at Wizard of Vegas.