Combinations and permutations with cards

Order matters in some counts and not in others. A permutation counts arrangements, so ABC and BAC are different. A combination counts groups, so those two arrangements are the same hand. Card games need both ideas, and they punish you if you use the wrong one.

This is for adults 18 and older. The arithmetic is exact when the deck is fair and well defined. It is not a staking plan. Laws on wagering differ by place.

Permutations: the deal in order

Suppose you care about the sequence of cards off the top. The first card can be any of 52. The second can be any of the 51 that remain. The third can be any of 50. The number of ordered three-card deals is 52 times 51 times 50, which is 132,600.

That number is a permutation. It fits questions like "how many ways can the next three cards arrive in a specific order." It is the wrong number if the rules only care which three cards you hold, not the order the dealer slid them.

Independent events are a different topic. A permutation can still describe dependent draws, because each factor uses the shrinking deck. Multiplying 1/52 by 1/52 by 1/52 would be the independent, with-replacement version. Ordinary dealing is without replacement.

Combinations: the hand as a group

A five-card poker hand does not care which card arrived first. The number of distinct hands is the combination C(52, 5). You can compute it as 52 times 51 times 50 times 49 times 48, divided by 5 times 4 times 3 times 2 times 1. The division strips out the order, because five cards can be arranged in 120 ways.

The result is 2,598,960. That is the count of different five-card hands from a 52-card deck. It is a famous figure because it is the denominator for poker odds. Texas hold'em hand rankings tell you which groups beat which. Combinations tell you how many groups of each kind exist. Rarity and rank are related and not identical. A hand can be rare and still sit in the middle of the ranking if the rules say so.

A smaller check you can do by hand

How many two-card combinations are in a 52-card deck? Ordered deals: 52 times 51. Unordered hands: divide by 2, because each pair was counted twice. That is 1,326. If someone quotes a two-card chance and their denominator is 52 times 51, they are counting order. Ask whether the game cares about order before you accept the fraction.

The same slip appears in "number of ways to be dealt two aces." Ordered: 4 choices then 3, which is 12. Unordered: 12 divided by 2, which is 6. There are C(4, 2) equals 6 ace pairs. Both counts can lead to the same probability if the rest of the fraction matches them. Mixing an unordered numerator with an ordered denominator is how people invent a chance.

What the count cannot do

A correct count is a description of a well-shuffled deck under stated rules. It does not say the next session will resemble the long-run share of those hands. Variance sits on top of any exact fraction. It also does not say a rare hand is a good purchase. The payout has to be set beside the fraction. A rare event with a thin payout is an expensive ticket, not a hidden bargain.

If you use these counts, write the question first. "Ordered or not?" "With replacement or without?" "Which deck, and which cards are already gone?" Then do the arithmetic twice. A clean fraction will not protect a budget. If you are staking money, the limit still comes from what you can lose, not from how elegant the denominator looked.

Frequently asked questions

When do I divide by the arrangements? When the game treats two orders of the same cards as the same outcome. Poker hands are the usual case. A specific order of community cards may still be a permutation problem.

Is 2,598,960 the number of hold'em starting hands? No. That figure is five-card combinations from 52 cards. A two-card starting hand is a much smaller combination, 1,326 if suits make hands distinct.

Does a correct count remove the house margin? No. It tells you how often a hand appears. The price is a separate sentence, and the long-run average can still be negative.