1,326 starting hands, 169 types: counting Hold'em combos

Two cards from a 52-card deck can be dealt in 1,326 ways, and a hand chart has far fewer entries than that. The difference is suits, which matter for flushes but not for the rank of your starting cards. This post counts the 1,326, shows how they fold into 169 types, and then shows how your own cards remove some of your opponent's possible hands.

Where 1,326 comes from

The first card can be any of 52. The second can be any of the remaining 51. That is 52 × 51 = 2,652 ordered deals, but the order you receive the cards in does not matter, so each hand has been counted twice. Divide by two:

C(52, 2) = 52 × 51 ÷ 2 = 1,326.

Holding the A♠ and then the K♥ is the same hand as holding the K♥ and then the A♠.

Folding 1,326 into 169 types

Two hands are the same type if they have the same ranks and are either both suited or both offsuit (pairs are neither). There are three kinds, and each has its own count of combinations.

1. Pairs: 13 types, 6 combinations each. Choose 2 of the 4 suits of a rank: C(4, 2) = 6. For aces that is A♠A♥, A♠A♦, A♠A♣, A♥A♦, A♥A♣ and A♦A♣. Total: 13 × 6 = 78.
2. Suited non-pairs: 78 types, 4 combinations each. Choose two different ranks, C(13, 2) = 78, and both cards must share one of the 4 suits. Total: 78 × 4 = 312.
3. Offsuit non-pairs: 78 types, 12 combinations each. The same 78 rank pairs have 4 × 4 = 16 suit arrangements, and 4 of them are suited, so 16 − 4 = 12. Total: 78 × 12 = 936.

Types: 13 + 78 + 78 = 169. Combinations: 78 + 312 + 936 = 1,326, so nothing is missing and nothing is counted twice.

What the counts tell you

Each pair type has 6 combinations, each suited type has 4 and each offsuit type has 12. So you are dealt a particular pair, such as kings, 6 ÷ 1,326 = 0.45% of the time, about 1 in 221. A particular suited hand, such as ace-king suited, comes 4 ÷ 1,326 = 0.30% of the time. A particular offsuit hand, such as ace-king offsuit, comes 12 ÷ 1,326 = 0.90% of the time.

That gives these ratios, which you can check from the totals:

• Ace-king offsuit is three times as common as ace-king suited (12 against 4).
• Ace-king offsuit is twice as common as a given pair (12 against 6).
• Any pair at all is 78 ÷ 1,326 = 5.9% of deals, about 1 in 17.

The suited or offsuit post uses the same 4-against-12 split to work out how much a matching suit adds.

Card removal: your cards change theirs

The counts above are for a fresh deck. Once you have looked at your two cards, your opponent is dealt from the other 50, so there are C(50, 2) = 1,225 hands they can hold. The 101 hands that disappeared are the ones that contain either of your cards: 51 + 51 − 1, because your own hand is counted in both lists.

Some types lose a lot more than others.

• You hold A♠K♥. Three aces remain, so the opponent has C(3, 2) = 3 combinations of aces instead of 6. Kings are the same, 3 instead of 6. Ace-king itself drops from 16 combinations to 3 × 3 = 9, of which 2 are suited (A♦K♦ and A♣K♣) and 7 offsuit.
• You hold A♠A♥. Only two aces remain, so the opponent has C(2, 2) = 1 combination of aces instead of 6. Ace-king falls from 16 combinations to 2 × 4 = 8. Kings are untouched at 6.

The effect is that a hand sharing a rank with yours gets rarer. When you hold an ace, an opponent with two aces is much less likely than the fresh-deck count suggests, and an opponent with ace-king is less likely too, since there are fewer aces for them to hold. It also matters when your hand is the one being dominated, as in the post on dominated hands: the ace in your hand is one fewer for them.

Reading the numbers

Treating the three kinds the same is the usual mistake. A chart box for ace-king offsuit and a chart box for pocket kings look equal in size, yet the first covers twice as many deals. Counting combinations, not boxes, is how you compare the share of hands a range covers.

1. Count pairs at 6, suited at 4, offsuit at 12. Add up the types in a range with those weights and divide by 1,326 to get the share of all deals.
2. Subtract for your own cards. Pairs of a rank you hold drop to 3 combinations if you hold one card of it and to 1 if you hold two.
3. Divide by 1,225, not 1,326, once your two cards are known, if you want your opponent's chance of holding a given hand.

For example, if you hold A♠K♥, the chance an opponent holds a pair of aces is 3 ÷ 1,225 = 0.24%, against 6 ÷ 1,326 = 0.45% from a fresh deck.

The "Hand charts" lesson in Holdem Coach: Poker Trainer shows at a glance which two-card hands to play from each seat, and the "Which hands to play" lesson asks whether your two cards are worth raising with when nobody has bet yet. Knowing how many deals each type represents tells you how often it comes up.