[pococurante]

I tried counting the possible hands, but it's not working.

Suppose you start with 3 of a kind in your hand. For each rank, there are 4 possibilities (trip aces without Ah, trip aces without As, etc.). So there are 52 ways to hold trips in your hand.

Now add a fourth card to your hand. There are 49 possibilities, so there are now 2548 ways to have them.

Now add a fifth card. 48 cards are there to choose from, meaning that you have 122,304 ways to have 3 of a kind in your hand. (You might also have quads or a full house right now... anything that includes 3 of a kind is possible.)

Can someone tell me why poker odds sites say there are 54912 trips, 624 quads, and 3744 full houses... for a total of 59280 ways to have 3 of a kind in a 5 card hand?

No point in doing the math if your starting numbers are all wrong.