Range combinations and card removal

Ace-king names a hand class. Ac Kd names one exact two-card holding. A range chart puts hand classes into equal-sized squares, but those squares contain different numbers of combinations. Counting the cards inside each square gives you a firmer starting point for reviewing a hand.

This lesson uses a standard 52-card deck and two private cards in Hold’em. You will count pocket pairs, separate suited and offsuit hands, then remove the exact cards known to be unavailable. The workbench lists every surviving holding so you can check the arithmetic yourself.

Ace: “AA and AK each take one square. Should I give them the same count?”

Aurora: “Open the squares. AA contains six exact holdings before removals; AK contains sixteen when we include all suits.”

Start with the exact two cards

An exact holding identifies both ranks and both suits. We use c for clubs, d for diamonds, h for hearts and s for spades. The rank letters are A, K, Q, J and T, with T meaning ten. Thus Ah Kd means the ace of hearts and the king of diamonds.

The order of the two cards does not create another holding. Ac Kd and Kd Ac describe the same physical pair. A counter that treats them as separate would double its result. The workbench forms each unordered pair of remaining cards once, then checks whether it matches your selected ranks.

A hand class groups exact holdings under one label. AA is a pocket-pair class. AKs is suited ace-king, AKo is offsuit ace-king, and AK in this lesson includes both. Keep that last convention visible when you copy a range from another source, because its notation may need clarification.

Why a pocket pair has six combinations

There are four aces in a full deck. Choosing a first ace gives four choices and choosing a second gives three. That produces twelve ordered selections. Each physical holding appears twice, once in each order, so dividing by two leaves six AA combinations.

Those six holdings are Ac Ad, Ac Ah, Ac As, Ad Ah, Ad As and Ah As. Listing them is a useful check when a formula feels too abstract. Each contains two different cards, and no row merely reverses a row already on the list.

If n cards of a rank remain available, its pocket-pair count is n x (n – 1) / 2. Four available cards give six combinations; three give three; two give one. With one or zero available cards, the count is zero because you cannot build a pair from a single card.

Every pocket pair uses two different suits. There is only one ace of hearts in the deck, so Ah Ah is impossible. Selecting AA with the same-suit filter is a valid question with an answer of zero. Selecting different suits returns every surviving AA combination.

Count different ranks and split the suits

For AK with no cards removed, each of four aces can go with each of four kings. Four times four gives sixteen combinations. There is no division by two in this calculation because the two roles are already distinct: choose the ace, then choose the king.

Four holdings share a suit: Ac Kc, Ad Kd, Ah Kh and As Ks. The other twelve have different suits. This gives the familiar unblocked counts of six for a pair, four for a suited non-pair class and twelve for an offsuit non-pair class.

The same starting counts apply to QJ, 76 or any other pair of different ranks. Hand strength does not change how many physical cards of each rank exist. Strategy enters later when you decide which holdings belong in a range and how often those holdings reach a particular decision.

Remove cards by identity

Suppose Ah is in your hand. It cannot also be in an opponent’s hand. Remove every proposed holding that contains Ah. Three aces remain, so AA falls to three combinations. AK falls to twelve because three available aces can pair with four available kings.

For suited ace-king, Ah removes Ah Kh. Ac Kc, Ad Kd and As Ks survive, giving three suited combinations. Nine offsuit combinations survive. One known ace therefore changes these classes by different proportions; it does not simply halve every hand containing an ace.

Now make both Ah and Kd unavailable. There are three aces and three kings left, so all-suit AK has nine combinations. The suited survivors are Ac Kc and As Ks. The heart combination lost its ace, while the diamond combination lost its king. Seven remaining combinations are offsuit.

Change the known cards to Ah Kh. The unrestricted count is still three times three, or nine. This time only the heart suited combination is broken, leaving Ac Kc, Ad Kd and As Ks. The split is three suited and six offsuit. Exact suits matter even when the rank counts look identical.

Ace: “I removed an ace and a king in both examples. Why did the suited count change?”

Aurora: “Ah Kd breaks two matching-suit pairs. Ah Kh breaks just the heart pair. Name the actual cards before counting.”

Use the workbench as an inventory

Choose the two ranks you want to inspect. Select all suits, same suit or different suits. The known-card box starts with Ah Kd. Submit that example with A and K selected, and check that the result lists nine exact holdings, including two suited holdings and seven offsuit holdings.

Enter known cards with spaces between them. Codes ignore case, so ah kd and Ah Kd mean the same thing. Ten can be written as T or 10: 10h becomes Th before duplicate checking. Entering both 10h and th is rejected because they name one physical card twice.

A blank known-card box means no cards have been removed. You may enter any set of zero to fifty-two distinct cards for a deck-inventory exercise. The counter does not check whether that set could be the board and your hand in a normal deal. Use actual known cards when reviewing a hand history.

The result always shows all-suit, same-suit and different-suit counts for the selected ranks, then lists the holdings matching your filter. A different-rank class has at most sixteen physical holdings, so the complete list fits on the page. A result of zero is possible and does not imply an input error.

Changing any input clears the old result. Calculate again to see the new inventory. Reset example restores A, K, all suits and Ah Kd, then returns focus to the first rank. If JavaScript is unavailable, the worked lists and exercises in the article remain readable.

Decide what is actually known

For an ordinary heads-up review, your two hole cards and the exposed board are known to you and unavailable to the opponent. Enter each once. On the flop that commonly means five cards; on the river it commonly means seven. The exact cards, including suits, determine the surviving inventory.

Unknown folded cards do not let you delete a particular ace or heart. Those cards have physical identities, but you have not observed them. Choosing their identities for the counter would add information that your decision did not have. Keep unseen cards unknown unless the exercise explicitly supplies them.

Exposed dead cards can belong in the unavailable set when their identities are established. Keep the source of that information in your study note. The workbench accepts more than seven known cards to support inventory checks; that capacity is not a claim that every entered scenario is a standard Hold’em deal.

Do not remove the opponent’s proposed cards while asking whether those cards could be their holding. If you enter Ac Kc as unavailable, Ac Kc cannot appear among the survivors. A revealed showdown can answer a different question about future unseen cards, so define the question before choosing removals.

Avoid overlap when combining classes

With no removals, AKs plus AKo contains sixteen distinct holdings. The two subsets do not overlap. AK plus AKs also contains sixteen distinct holdings, because every suited ace-king holding is already included in AK. Adding sixteen and four would count those four holdings twice.

A range is a set of exact holdings, so repeated labels do not add new cards to it. Before summing class counts, expand or otherwise identify overlaps. The workbench inspects one rank pair at a time and does not combine a pasted range expression for you.

For a small example, a range containing AA and AK has twenty-two unblocked combinations. These classes cannot overlap because one contains two aces and the other contains an ace and a king. With Ah Kd removed, AA contributes three and AK contributes nine, for twelve surviving holdings.

Give action weights their own step

Counting a legal holding does not tell you how often an opponent reaches the decision with it. A player may open one class often, call another only occasionally and fold others earlier. Those action frequencies attach weights to the physical inventory you have just counted.

Suppose, purely as a model, AA and AK are the only classes and every exact holding has equal weight. Before removals, AA accounts for six of twenty-two holdings, or about 27.27%. That proportion follows from the assumed range and equal weighting. It is not a measured opponent tendency.

Now suppose each AA combination reaches the decision with weight 1 and each AK combination with weight 0.5. The physical inventory still contains six AA and sixteen AK holdings. Their weighted masses are six and eight, so AA accounts for six of fourteen weighted units, about 42.86%.

A half-weight holding remains one physical holding. The weight says how much it contributes to this constructed range after an action assumption. After removals, delete incompatible holdings first and then sum the weights on survivors. Some models also assign different weights to different suits on a given board.

The workbench deliberately supplies no weights. If you want to estimate a betting range, write down the earlier actions, the assumed frequencies and why you chose them. That makes disagreements about the model visible instead of hiding them inside a precise-looking combination count.

A blocker affects every category it touches

A known card removes every incompatible holding in the range being considered. Some may be value bets; others may be missed draws used as bluffs. Whether a removal helps a proposed bluff or call depends on both groups and on their action weights.

On a board with a possible flush, a suited card in your hand may remove possible flush holdings. It can also remove hands from other parts of the range, including potential bluffs. Naming the card alone does not establish which effect dominates. List the affected exact holdings in each category.

A smaller value inventory does not establish a sufficient bluff frequency. You still need a model for which bluffs take this line, the bet size and the value hands that remain. Legal possibilities and betting choices answer different questions, even when both are described with combinations.

Ace: “I can cross out two value holdings. Does that make this a call?”

Aurora: “Keep the crossings. Then check the bluff holdings and their weights. The removal count is only one part of the decision.”

Check your count against the whole deck

A full deck contains 52 x 51 / 2 = 1,326 unordered two-card holdings. The thirteen pair classes contribute thirteen times six, or seventy-eight. There are seventy-eight different-rank combinations of ranks, each supplying four suited and twelve offsuit holdings.

The total is 78 + 312 + 936 = 1,326. These holdings fit into 169 familiar chart classes: thirteen pairs, seventy-eight suited classes and seventy-eight offsuit classes. Counting the chart squares as equally likely exact holdings would discard the different sizes of those classes.

With k distinct known cards removed, 52 – k cards remain. The whole-deck two-card inventory is (52 – k) x (51 – k) / 2 when at least two cards remain, and zero otherwise. Seven known cards leave forty-five cards and 990 holdings across all rank classes.

That total is a useful independent check. If you count every rank class after the same removals, the class totals should sum to the whole-deck inventory. A single rank-pair result such as nine AK holdings is only a subset of that total.

Work a removal sequence before submitting

Start with QQ, all suits and no known cards. Predict six. Enter Qh and predict three; add Qd and predict one, Qc Qs. Add Qc and predict zero. Each new known card can remove raw combinations or leave the count unchanged. It cannot create a new physical holding.

Next select A and K with Ah Kd known. Predict nine all-suit holdings, two same-suit and seven different-suit. Change Kd to Kh and predict nine, three and six. Read the complete list to identify which suited holding returned and which offsuit holdings changed.

For a final check, keep A and K selected but remove only 7c 8d. All sixteen AK holdings survive because neither known card belongs to the selected ranks. Known cards can reduce the deck size without changing a particular class. The inventory result should make that distinction clear.

Record the selected class, known cards, surviving exact holdings and any later weighting assumptions in separate fields of your hand review. You can then correct a card transcription without silently changing the action model, or revise the model without pretending the deck count changed.

For adults of legal playing age. Use this off-table lesson for study and keep gambling within affordable limits.

Back to top