Combinatorics lab
A pocket pair is six hands. Ace-king is sixteen — four suited, twelve offsuit. Hold one ace and the opponent’s aces drop from six combinations to three. This is the arithmetic behind blockers and ranges, shown one card at a time.
Your hole cards, the board, anything exposed. Up to seven. Every count below is recomputed as you add or remove them.
Ace-King
9 / 16 combinations live| K | K | K | K | |
|---|---|---|---|---|
| A | suited | blocked | offsuit | offsuit |
| A | offsuit | blocked | offsuit | offsuit |
| A | offsuit | blocked | suited | offsuit |
| A | blocked | blocked | blocked | blocked |
- Suited combinations4one per suit
- Offsuit combinations124 × 3
- All combinations164 × 4
- As you can see1
- Ks you can see1
- Live combinations93 × 3One high card times one low card; every pairing is a distinct combination.
Blockers at a glance
How the cards you can see change what an opponent can hold.
| Hand | Combos | Now | Removed |
|---|---|---|---|
| AA | 6 | 3 | 50% |
| KK | 6 | 3 | 50% |
| 6 | 6 | 0% | |
| AKs | 4 | 2 | 50% |
| AKo | 12 | 7 | 42% |
| AQs | 4 | 3 | 25% |
| AQo | 12 | 9 | 25% |
| KQs | 4 | 3 | 25% |
| JTs | 4 | 4 | 0% |
| 77 | 6 | 6 | 0% |
Card removal across a range
209.9 of 265.2 combos remain · 20.9% removedGreen: untouched. Red: combinations removed by the cards you can see; the deeper the red, the larger the share. Each cell reads live / total.
Most affected
- AKo12 → 7 (−5)
- AQo12 → 9 (−3)
- AJo12 → 9 (−3)
- ATo12 → 9 (−3)
- A9o12 → 9 (−3)
- A8o12 → 9 (−3)
- A7o12 → 9 (−3)
- A5o12 → 9 (−3)
This is the whole idea of a blocker: a card in your hand or on the board is a card the opponent cannot have, and every combination that needed it disappears from their range. Sample ranges are educational examples, ordered by equity against a random hand.
Why 6, 4 and 12
A pocket pair chooses two suits from four: C(4, 2) = 6. A suited hand needs both cards in the same suit: one combination per suit, 4. An offsuit hand has four choices for the first card’s suit and three for the second: 12. Any specific unpaired hand is therefore 16 combinations, and offsuit outnumbers suited three to one.
Card removal is just multiplication with smaller numbers. With one ace visible, aces are C(3, 2) = 3 and ace-king is 3 × 4 = 12. With an ace and a king visible, ace-king is 3 × 3 = 9.
Why it matters
Every equity-against-a-range number on this site is a combination-weighted average, so the counts here are the weights. A hand that blocks half of an opponent’s strongest holdings changes their range far more than it changes yours.
See the effect on equity in Hand vs range, where the same removal runs automatically, and learn the counting in the combinations lesson.