The odds quoted in the OP are obviously the odds of being dealt any three of a kind first 3 cards.
You will never work out the odds of three of a kind being your best hand with simple permutations. You have to eliminate every combination that includes a pair because you now have a full house. Similarly any hand where the other 4 cards contibuted to a straight, flush or straight flush has to be eliminated.
Of the 133,784,560 possible 7 card hands 6,461,620 have at best 3 of a kind and 1/13 of these would be aces.
A good table of probabilities is here.