Find the number of five-card hands, dealt from a standard 52-card deck, that contain three of a kind (three cards of one denomination, a fourth card of a different denomination, and a fifth card of a third different denomination);

There are 13 different denominations to choose the denomination for the three of a kind, multiply by 4 choose 3 ways to determine the suits of those three cards, multiplied by 12 choose 2 denominations for the fourth and fifth cards times 4 ways to choose the suit of each of those cards. Altogether this gives 13 * 4C3 * 12C2 * 4 * 4 = 54,912.

Is this correct? (Ace is being counted as both high and low card)

I apologize for the combination notation but I haven't been able to get the math tags working properly again.