A 5-card hand is randomly drawn from a standard 52-card deck. What is the probability of getting "three of a k?

First consider how many possible combinations of 5 card hands possible:
52C5 = 2598960

Now consider the combinations with 3 of a kind:
13C1 : which rank will the three of a kind be,
4C3 : of the rank chosen, which 3 cards will be picked
49C2: of the remaining 49 cards, pick two more to complete the hand,

so 13C1 * 4C3 * 49C2 = 61152

Your probability is: 61152/2598960 = 0.0235294 = 2.35%

Note, hands like full house and four of a kind are also considered for the probability calculated above. If you you only want the probability of three of a kind, (four of a kind, full house excluded) then you have to subtract the combinations for four of kind and full house from 61152.

For full house: 13C2 * 4C3 * 4C2 = 1872 combinations
(choose two ranks, one for three of a kind and one for pair, choose which 3 for three of a kind and choose which 2 for the pair)

For four of a kind: 13C1 * 4C4 * 48C1 = 624
(by similar arguments as the two above)

61152 - 1872 - 624 = 58656
So the probability of three of a kind ONLY is: 58656/2598960 = 0.022569 = 2.2569%