If you are dealt 5 cards, the probability of getting 3 of a kind:
You choose 3 cards from 4 of the same. i.e. 3 aces from the 4 aces, 3 kinds from the 4 kings, etc. There are 13 suits.
So, we have 13C(4,3)
But, we have to choose 2 cards from the remianing 48, so we have:
There are C(52,5)=2,598,960 different possible combos, so the probability of getting a 3-of-a-kind is
But...one little thing to mention. This also includes the fullhouse. Afterall, a fullhouse is a fancy three of a kind. To cut to the quick, there are 3744 different fullhouses. You can subtract this if you wish.
Try to count up some of others. Get out a deck and think it through. It's rather fun.