How many different 3 of a kinds are there (3 cards of one value 2 of any other value) in a 5 card hand? (Give the exact number)
We construct the hand step by step, keeping in mind that we don't want to accidentally include the better hands of four of a kind or full house.
First choose the value that will appear three times: this can be done in 13 ways.
Now choose the suits these cards will have: we have three cards and can choose between four suits, so this can be done in "4 choose 3", or 4 ways.
Now choose the values of the other two cards: we are now choosing from only twelve cards, and we don't want to choose two of the same card. This can be done in exactly "12 choose 2", or 66 ways.
Now, finally, choose the suit of these last two cards: each one can have any of the four suits, so this can be done in 4*4=16 ways.
Combining all of this using the multiplication principle, we see that there are exactly 13*4*66*4*4=54,912 different three of a kind hands.
It's instructive to consider some common mistakes as well. For example, in the last step, we have two cards and we want to choose their suits. Why can't this be done in "4 choose 2", or 6, ways? Well, for one, this doesn't take into account order (king of diamonds and 6 of spades isn't recognized as different from king of spades and 6 of diamonds, for example), so we have to at least double our number and get 12. But the "choose" operation also doesn't allow duplicates, so something like king of hearts and 6 of hearts isn't even an option. There are 4 ways for the two cards to have the same suit, so we finally correct for our "mistake" (it's not really a mistake if you're aware of its limitations and correct for it, however) and end up with the correct result of 16.
There are always slight variants too. After we have chosen our three initial cards and their suits, we can now think of choosing two cards out of the remaining 48 (we can't use the three we've chosen or the fourth one of the same value, or else we'd have a four of a kind). There are 1128 ways to do this, but among these ways are hands that give us a full house. That happens when the two cards selected have the same value: there are 12 ways to pick this value and "4 choose 2" ways to assign them suits. We subtract these out, and indeed 1128-12*6=1056, which is exactly the result we get when we carry out the calculation the first way: 66*4*4=1056