Hey guys,

lets imagine we have a really nitty 4better and are safe to assume he is only 4betting AAxx.

We call (with whatever hand) and see a disconnected flop with a flushdraw like Qs7s3d. Given we know villain is only 4betting AAxx how likely is he to have flopped a flushdraw here?

My plan was to first figure out how often he has AAxx ss, ds, ts, rainbow and proceed from there. However i encountered some problems on the road.

The amount of all (AAxx)-combinations should be given by: c(4,2)*c(48,2)=3!*(48*47/2)=6768

For (AAxx)ds-combinations: c(4,2)*c(12,1)*c(12,1)=864
(AAxx)ss: c(4,2)*c(24,1)*c(24,1)=3456
(AAxx)ts: c(4,2)*c(12,2)=396
(AAxx)r: c(4,2)*c(12,1)*c(12,1)=864

So if i sum up all of these i should have my 6768 possible (AAxx)-combos. However i am missing a ton and have no clue where they are. I suppose (AAxx)ts could look like: c(4,2)*c(2,1)*c(12,2) since i have 2 colours to choose from but that still leaves a bunch of combinations missing. Any ideas or help would be appreciated!

You are close. You have all the double suited and rainbows. You are on right track with three of suit. Think of the single suited this way and it might help,

Single suited to pair
Single suited not to pair
Three of suit

I'm on my phone and doing this on the fly, but it is something like this for AAxx not trips or quads.

0864 double suited
3456 single suited to pair
0792 single suited not to pair
0792 three of suit
0864 rainbow

6768 total

As a side note, the single suited to pair can be suited to the high card or the low card, but I am not sure that matter here in the case of aces since a pair of aces is also the highest card of suit. But for example, 77xx single suited to pair could be single suited to a card higher than a 7 or lower than a 7.

You aren't factoring in the removal effects of the board cards and you are also discarding AAA and AAAA hands (the latter may be intentional)

On a board of Qs7s3d there are:

6121 combos of AA (6*45c2 AA + 4*45 AAA + 1 AAAA)

Working upwards
30 combos of AAA with a spade flush draw (3*10)
1080 combos of AA with an A high spade flush draw and exactly 2 spades (3*10 + 3*10*35)
1215 combos of AA with an A high spade flush draw (3*10 + 3*10*35 + 3*10c2)
1350 combos of AA with a spade flush draw (3*10 + 3*10*35 + 3*10c2 + 3*10c2)

The 3 in the above formulas comes from 3 choose 1 (which is the number of ways to make AA with As, AA without As, or AAA with As), the 10 comes from 10 remaining spades in the deck not counting the As, Qs, or 7s, and the 35 comes from 35 non spade, non A, non board cards left in the deck (52 - 3 board cards - 4 Aces - 10 other spades).


For your question about the total combos of AAxx (with no board cards):
You are correct about the number of rainbow and double suited combos being 864.
Triple suited you need to multiply by 2 as you stated towards the bottom so it's 4c2 * 12c2 * 2 since there are 2 different aces you can be triple suited to, so it's 792 combos.

For single suited there are 4c2 * 12c1 * 24c1 * 2 = 3456 ways to be suited to an A (6 ways to make AA, 12 choices for suited card, 24 choices for unsuited card, and 2 different aces to be suited to). And to be single suited not to an A there are 4c2 * 12c2 * 2 = 792 combos (6 ways for AA, 12 choose 2 suited cards, and 2 opposite suits of the AA).

This is again excluding AAA (of which there are 4*12*3 = 144 suited combos and 4*12 = 48 unsuited combos) and AAAA (one rainbow combo).

So what's the answer guys?

Shove with any ds connectors bro.

If this is the question and since he is nitty and not 4-betting AAAx, then I think the answer is about 23% If we talking 4 card flush draw.

Of the 6,768 possible AAxx if we are looking for the suited spades, then that is half the double suited hands, a quarter of the single suited and a quarter of the triple suited hands. So 1,548 divided by 6,768 equals 22.87%.

It is the weekend and there is beer involved so this might be wrong, but I'd bet the next round of beers it is right.