The name of that sort of problem in probability is 'draw without replacement'.
When you do this sort of problem, it's easier to first figure out the probability of not getting what you want, then subtracting that from 100% to get the probability you're interested in.
First, we'll find probability of finding H alone. We start by finding the probability of having a hand without H. If we draw 5 cards, then the probability of the first one not being H is 39/40, then, if that happened, the probability of the second one not being H is 38/39. We want the probability of all draws not being H, so we multiply the individual probabilities with each other. The probability of all 5 cards not being H is
(39/40) * (38/39) * (37/38) * (36/37) * (35/36) if you multiply that out, you get .875, so that's a 87.5% chance of not getting H in a hand of 5, subtract the percentage from 100 and you have 12.5% chance of getting that H. If you do the same thing with 6 cards, you get 85% of not and 15% of getting it.
Did you follow all of that? I hope so, because combinations of 2 cards will get more complicated.
Doing the same thing we did for H, we can find that the probability of getting at least 1 A is 33.75%. The probability of getting at least 1 B is the same, but getting at least one of each together is a little harder.
To find the probability of drawing an opening hand with A and B
(P[A and B]), we need to find 3 probabilities first, the probability of drawing just A but not B
(P[A and ~B]), the probability of drawing just B but not A
(P[~A and B]), and the probability of drawing neither A or B
(P[~A and ~B]).
For P[A and ~B], take P[A] * P[~B]. That's
0.3375 * 0.6625 = .2235.
(NOTE: This is technically incorrect. It's an approximation that works okay when A and B are a lot less than the total number of cards)
Since the number of As is the same as the number of Bs, P[~A and B] is the same.
The neither part is similar to the 'not H' problem. Same as last time we do
total-cards-in-deck. Again for a hand of 5 cards:
(34/40) * (33/39) * (32/38) * (31/37) * (30/36) gives you 0.4228 for drawing neither A or B.
Then we do P[~A and ~B] + P[~A and B] + P[A and ~B] = 0.4228 + 0.2235 + 0.2235 = 0.8698. That's the probability of drawing 5 cards without the A and B combo. So, you'll have a 13.02% chance of having that combo.
I hope that's enough info for you to calculate the other combinations you're interested in.