Combo Counting — Narrowing Opponent Ranges with Mathematics

🔢 Combination Basics

Every possible two-card starting hand in Texas Hold'em can be counted using combinatorics. Knowing the exact number of combinations for each hand type lets you quantify ranges precisely rather than thinking in vague terms like "he probably has a big pair."

The number of ways to choose 2 cards from a set of \(n\) cards is given by the combination formula:

$$C(n, 2) = \frac{n!}{2!(n-2)!} = \frac{n(n-1)}{2}$$

The Three Core Cases

Pocket Pairs (e.g., AA, KK, 77): There are 4 cards of each rank in the deck. The number of ways to choose 2 from 4:

$$C(4, 2) = \frac{4 \times 3}{2} = 6 \text{ combos}$$

Suited Hands (e.g., AKs, JTs): There are 4 suits, and a suited hand uses two cards of the same suit. There are exactly 4 suit combinations:

$$\text{Suited combos} = 4$$

Offsuit Hands (e.g., AKo, QJo): All combinations of two different ranks where the suits differ. Total two-rank combinations = \(4 \times 4 = 16\). Subtract the 4 suited combos:

$$\text{Offsuit combos} = 16 - 4 = 12$$

🚫 Combos with Blockers

When you hold a card (or when board cards are visible), those cards are removed from the deck and opponent's possible holdings are reduced. This is the concept of blockers.

If you hold one card of a given rank, only 3 cards of that rank remain. The formula for a pocket pair when you block one of its cards:

$$C(3, 2) = \frac{3 \times 2}{2} = 3 \text{ combos remaining}$$

If you hold two cards of different ranks (e.g., AK), you reduce both AA and KK combos, and also reduce the number of AK combos the opponent can hold:

• You hold A: opponent can only have \(C(3,2) = 3\) combos of AA (not 6)
• You hold K: opponent can only have \(C(3,2) = 3\) combos of KK (not 6)
• You hold AK: opponent's AK combos = \(3 \times 3 = 9\) (not 16), since only 3 aces and 3 kings remain

General Blocker Formula

For any unpaired hand where you hold \(b_1\) blockers to rank 1 and \(b_2\) blockers to rank 2:

$$\text{Remaining combos} = (4 - b_1) \times (4 - b_2)$$

For a pocket pair where you hold \(b\) blockers to that rank:

$$\text{Remaining combos} = C(4 - b, 2)$$

📊 Counting a Range Step by Step

To count the total combos in a range, follow this process:

1. List all hand types in the range (AA, KK, AKs, AKo, etc.)
2. Count combos for each hand type, adjusted for board cards and your hole cards.
3. Sum all combos to get the total range size.
4. Divide subsets (e.g., value hands, draws, bluffs) by total to get percentages.

Example: Counting a Simple Value Range

Villain 3-bets and you assign them: {AA, KK, QQ, AKs, AKo}. Board: K♠7♦2♣. You hold A♠J♦.

• AA: you block one ace → \(C(3,2) = 3\) combos
• KK: board has K♠, you hold no king → \(C(3,2) = 3\) combos
• QQ: no blockers → \(C(4,2) = 6\) combos
• AKs: you block A, board blocks K → \((4-1)\times(4-1) = 9\) total AK, of which suited = 2 (A♥K♥ and A♦K♦ remain with live suits, since A♠ is yours and K♠ is on board) → 2 suited combos
• AKo: \(9 - 2 = 7\) offsuit combos

Total: \(3 + 3 + 6 + 2 + 7 = 21\) combos. Villain has sets/overpairs (6 combos = 29%) and AK (9 combos = 43%) in their range.

🃏 Three Worked Examples

Example A: AA Combos on K72 Rainbow Board

You hold Q♠Q♦ on K♠7♥2♣. How many AA combos can villain have?

You hold no ace. Board has no ace. All 4 aces are live.

$$\text{AA combos} = C(4, 2) = 6$$

All 6 AA combos are possible. This is important when facing aggression on this dry board — villain's range includes 6 AA combos that completely dominate your QQ.

Example B: AK Combos with King on Board

Board: K♦9♥4♣. You hold A♠T♠. How many AK combos can villain have?

• Aces remaining: 4 − 1 (you hold A♠) = 3 aces
• Kings remaining: 4 − 1 (K♦ on board) = 3 kings

$$\text{AK combos} = 3 \times 3 = 9 \text{ total}$$

Of the 9 AK combos, suited AK combos: A♥K♥, A♦K♦, A♣K♣ = 3 (note A♠K♦ is not suited, and K♦ is on board so A♦K♦ is valid since you need A♦ and K♦ — but K♦ is on the board). Actually: K♦ is on the board, so no combo involving K♦ is available. Re-check: remaining kings are K♥, K♠, K♣. Suited AK with these: A♥K♥, A♣K♣ = 2 suited combos. Offsuit AK = 7 combos.

Example C: Flush Draw Combos on Two-Tone Board

Board: J♥8♥3♣. How many flush draw combos (nine-high flush draws, specifically K♥x♥) are possible?

Hearts remaining in deck: 13 total − 2 on board (J♥, 8♥) = 11 hearts. You hold Q♥ in your hand: 10 hearts remain for villain.

Number of two-card flush draw hands villain can hold (both hearts, not J♥ or 8♥):

$$C(10, 2) = \frac{10 \times 9}{2} = 45 \text{ heart-heart combos}$$

These 45 combos represent all possible heart flush draws villain can hold — a significant portion of any reasonable range. This is why wet boards require more caution than dry boards.

⚖️ Value-to-Bluff Ratio via Combos

Combo counting lets you assess whether a betting range is balanced. The optimal bluff-to-value ratio on the river is determined by the pot odds offered to the caller:

$$\text{Bluff combos} = \text{Value combos} \times \frac{\text{Bet}}{\text{Pot} + \text{Bet}}$$

For a pot-sized bet, the caller needs 50% equity to break even, so the bettor must have 50% value and 50% bluffs — equal numbers of each.

For a half-pot bet (caller needs 33% equity), the bettor needs:

$$\text{Bluff ratio} = \frac{0.5}{1.0} = 0.5 \quad \Rightarrow \quad \text{1 bluff per 2 value bets}$$

Example: You have 9 value combos (sets and two pair) on the river and bet 50% pot. The optimal number of bluff combos is \(9 \times 0.5 = 4.5 \approx 4\text{–}5\) combos. If you are bluffing with 12 combos alongside 9 value combos, you are over-bluffing and exploitable.

💡 Practical Usage

Combo counting at the table requires practice but can be simplified with these mental shortcuts:

• Memorize the base numbers: pairs = 6, suited = 4, offsuit = 12. These are your starting points for every count.
• Subtract for each blocker: each card you or the board holds reduces the relevant rank by 1. For pairs: 4→3 cards = 3 combos, 4→2 cards = 1 combo. For unpaired: multiply remaining cards of each rank.
• Focus on key hand categories rather than individual combos. How many sets are possible? How many two-pair combos? How many flush draws? Sum these buckets.
• Use combo ratios for quick decisions: if the board makes sets possible for villain and there are 9 set combos vs. 4 bluff combos in a likely range, the range is heavily value-weighted — call cautiously or fold against large bets.