Odds of two pairs

Two pair hands are defined by the rank combos of the pairs, the suit combos of the biggest and the smallest pair, and the kicker combos. The first pair can be bigger or smaller than the second pair. We're not interested in the arrangement of the pairs, so we divide the arranged rank combos by two.

n(two pair hands) = rank combos * suit combos(big pair) * suit combos(small pair) * kicker combos = (13*12/2) pair combos * 6 suit combos/pair combo * 6 suit combos/suit combo * 44 two pair hands/suit combo = 123 552 two pair hands

Aces up is two pair where the biggest pair is a pair of aces, kings up is two pair where the biggest pair is a pair of kings and so on. These categories might not be enough to decide if one hand beats another, since two players might have two pair in the same category.

n(threes up hands) = small pair combos * suit combos * kicker combos = 1 small pair * 36 suit combos/small pair * 44 threes up/suit combo = 1584 threes up

I just lumped together the suit combos here, because there are always 6 suit combos for each pair and these combos are independent from each other. 1584 is actually the number of possible hands for any combination of two specific pairs (1584 "fours and deuces hands", 1584 "aces and kings hands" and so on). The only pair that is smaller than threes, are deuces. Fours up can be either "fours and threes" or "fours and deuces", so there are twice as many fours-up hands as trees-up hands.