A deck of cards consists of 52 cards. When dealing a hand in Texas
Hold 'em every player gets two cards, and there are precisely 1326 combinations - "combos"
- of starting hands.
This is because the first card is any one of the 52 possible cards, and the second one is any one of the 51 remaining. We multiply 52*51 and get 2652 combinations, but in practise there's no difference between and , so we count them as the same hand, halving it from 2652 to 1326.
Knowing that there are 1326 combos of starting hands is not particularly interesting. What's of interest is the fact that understanding combos means we can infer some juicy stuff about what our opponents are likely to be doing. By developing reads and/or using analytical programs such as Hold'em Manager and PokerTracker, we can have some idea of how our opponent plays certain hands. By looking at the board, we can then make some intelligent guesses as to how likely he is to beat us.
To give a practical example, I want to start by showing what happens when we don't
$100NL, 6max hold 'em. We open preflop to $6 with
on the button, and a tight opponent makes it $13 in the small blind. The big blind folds.
We know, from past play with this opponent, that he only 3-bets his premium hands. This means QQ, KK, AA and AK. Because we know his habits so well, because we're in position and because we are getting a decent price on trying to flop a set and stack him, we call.
The flop is
Our opponent shoves his entire stack in. We know exactly what this means. Somehow, we know that means that he has either QQ, KK or AA, but not AK, because he would make a smaller bet with AK. Should we call?
On the surface, it may seem like there are more hands that beat us than that we can beat. He shoves AA, KK and QQ, and of those three, we can only beat AA. But this is misleading, and this is why combos are important:
Before the flop, the likelyhood of him having KK instead of AA was the same. But after the flop, this is no longer the case, because there's a K on the board
. That's one less king that he can have in his hand. Looking only at the pocket kings, before the flop he could have any of these hands:
... but now that the is right there, on the table, he can no longer have any of the combos that include that. Removing those, that leaves us with:
... only three.
We can then use the exact same argument with the pocket queens. Originally, there were six pocket queens that he could have, but now that a queen fell on the flop, there are only three left. What about aces? Well, there's no ace on the board, so all six combos of aces are still "available" to him, and we can therefore conclude this:
He is just as likely to have a hand that we beat as a hand that we don't beat (6 combos of each).
Because of the dead money already in the pot, we should therefore call.
Another, slightly more practical example, is when your opponent goes all-in on this board on the river in a big pot:
... and you have
Your opponent could be bluffing. He would also shove with a set.
The only hand that beats you is KQ. And you know (or I do, and am about to tell you) that there are 12 combos of KQ that he can have. Originally, there are 16 combos of KQ (king of hearts which each of the queens, king of spades which each of the queens, etc.), but you have one of the queens in your hand, so he can only have three queens for each of the four kings = 12.
12 combos beat you. How many combos do you beat? Well, the three sets that he could have flopped are "3 per set," i.e. three combos of jacks, three combos of tens and three combos of nines, for a total of 9 combos of sets
. Furthermore, he could have the lower end of the straight - 87 - which for the same reason as with KQ there are 12 of.
He can have 12 combos that beat you, and you beat 21 (9+12) combos. And that's not even accounting for the fact that he could be bluffing. Hence, you have an easy call.
So when doing hand analysis, we don't just have to look at "what hands" he can have, and how likely he is to play a certain hand, but also how likely that hand is combinatorically
. If he plays AA and KK the same way, and there's a king on the flop, he's twice as likely to have AA than KK - which could be very useful for you to know if you have a lower set.
Short summary of various types of hands (not accounting for cards that you know he can't have):
Pocket pair: 6 combos.
Suited connector (e.g. 87s): 4 combos
Offsuit connector (e.g. 87o): 12 combos
Any hand that doesn't have to be suited and isn't paired (e.g. "KQ"): 16 combos
This is tricker because it assumes stuff about our range. But let's make it easy for me and say "only pocket pairs or broadway cards"
Example flop 1 (dry flop): :Qd:
(three QQ + three 77 + three 22)
Two pair: None
(since Q7 isn't in his range to begin with, neither is 72 or Q2)
(6 KK + 6 AA)
Top pair: 48
(AQ, KQ, QJ and QT, 12 each)
Pairs between 77 and QQ: 24
(eights, nines, tens and jacks, each six combos)
Example flop 2 (wet flop):
Two pair: 9
(QJ; three queens can mix with three jacks = 3*3 = 9)
Top pair: 36
(AK, KQ, QT, 12 each)
Open-ended straight draws: 16
Gutshot straight draws: 16
Note: Flushdraws seem rare in this scenario. That's because they are in general. Lesson? Not everyone has a flushdraw every time the flop is twotone.
Discussion and questions are of course welcome.
Edit note: These examples are a bit cumbersome to put in - let me know if you want others added.