Module 3 · Lesson 15 of 39 · Free
Combinations of one hand: 6, 4 and 12
Before any card is seen, AK can be dealt in 16 ways and AA in only 6. Every card you can see takes some of those ways away: one ace on the board leaves AA with 3 combinations and AK with 12. Counting what is left tells you which hands an opponent is likely to hold.
The hand
The flop A♦ 8♣ 3♠. You hold J♣ J♦, and the Tight-aggressive player bets. Say he bets here with AA or AK and with nothing else.
Predict first
With the A♦ on the board, in how many ways can he hold AK?
- 16: the board does not change his cards
- 12: 3 aces left × 4 kings
- 9: 3 aces × 3 kings
Show the answer
Answer: 12: 3 aces left × 4 kings
Only 3 aces are left, and each can go with any of the 4 kings: 3 × 4 = 12. AA has only 3: the 3 aces left pair up in 3 × 2 ÷ 2 = 3 ways. So AK is 12 of his 15 combinations, 80%.
Working it out
Count a pair from the cards of its rank you cannot see: with n of them, n × (n − 1) ÷ 2. Four unseen give 6, three give 3, two give 1 and one gives none. An unpaired hand is the unseen cards of one rank times the unseen cards of the other: 4 × 4 = 16 before any card is seen, 3 × 4 = 12 with one ace gone. Suited combinations need both cards in one suit: with the A♦ gone, AKs keeps 3, in spades, hearts and clubs, and AKo keeps 12 − 3 = 9. Your own cards count too: holding J♣ J♦ leaves him only 1 combination of JJ.
A card that takes combinations away is a blocker. When you know he holds one of a few hands, weigh each by the combinations left: on this flop AK is four times as likely as AA, 12 against 3. Reading a whole range this way is the subject of the lesson Reading hands as distributions, in the Mathematics track; the counting is the same.
Formula
Pair: n × (n − 1) ÷ 2 with n unseen cards of its rank: 4 → 6, 3 → 3, 2 → 1. Unpaired: unseen of one rank × unseen of the other: 3 × 4 = 12
Check your understanding
You hold A♠ Q♠ and the flop is K♥ 7♦ 2♣. How many combinations of AK can an opponent hold?
Show the answer
Answer: 9
3 aces and 3 kings are unseen: 3 × 3 = 9.
Same cards. How many of those are AKs, with both cards of one suit?
Show the answer
Answer: 2
The A♠ and the K♥ are in sight, so only diamonds and clubs still have both cards unseen: 2.
Same cards. How many combinations of KK can he hold?
Show the answer
Answer: 3
3 kings are unseen: 3 × 2 ÷ 2 = 3.