Card games are games of chance with a built-in house edge in any house-banked format. No counting method, strategy or system changes the odds in the player’s favor over time, and most players lose money over time. Nobody should gamble money they cannot afford to lose. This article is about counting, not about winning.
Why counting matters
When all outcomes are equally likely, a probability is the number of favorable outcomes divided by the total number of outcomes, as laid out in the basics of probability. With a deck of cards the total is large, so listing outcomes by hand is impractical. Two counting tools, permutations and combinations, make it manageable.
Permutations: order matters
A permutation is an ordered arrangement. If you deal three cards from a 52-card deck and care about the order they arrive, there are:
52 × 51 × 50 = 132,600
The first card has 52 possibilities, the second has 51 (one card is gone), and the third has 50. In general, the number of ordered arrangements of k items chosen from n is n × (n − 1) × … for k terms, which is written n!/(n − k)!.
The number of ways to order the whole deck is 52!, which is about 8.07 × 10^67. That figure is why a properly shuffled deck is almost certainly in an order that has never existed before. It is a counting fact, and says nothing about whether any particular shuffle is fair, which is a separate question tested in the work on shuffling machines.
Combinations: order does not matter
Most card questions ignore order. A poker hand of ace-king-queen is the same hand however it was dealt. A combination counts selections without regard to order.
Start with the three-card example. Every group of three cards can be arranged in 3 × 2 × 1 = 6 ways. So the number of unordered three-card groups is:
132,600 ÷ 6 = 22,100
The general formula divides the permutation count by k!:
C(n, k) = n! ÷ (k! × (n − k)!)
This is read “n choose k.” You can confirm C(52, 3) = 22,100 from the formula.
Two-card hands
For a two-card starting hand, C(52, 2) = (52 × 51) ÷ 2 = 1,326.
This number is the denominator in several familiar questions:
- A specific pair, such as two aces: C(4, 2) = 6 combinations, so 6/1,326, about 0.45%.
- A two-card blackjack total of 21 (an ace plus a ten-value card): 4 aces × 16 ten-value cards (10, J, Q, K in four suits) = 64 hands. 64/1,326 is about 4.83%.
For the blackjack figure, multiplication rather than “choose” is correct because the two cards come from two different groups, the aces and the ten-value cards. Each pairing is a different hand. For deeper treatment of this game, see the discussion of blackjack mathematics.
Five-card hands
The number of distinct five-card hands from a 52-card deck is:
C(52, 5) = (52 × 51 × 50 × 49 × 48) ÷ (5 × 4 × 3 × 2 × 1) = 311,875,200 ÷ 120 = 2,598,960
Every poker hand frequency is some count divided by 2,598,960. The ranking order of hands, covered in hand rankings, follows from how rare each type is.
Some counts you can verify yourself:
- Royal flush. One for each suit, so 4 hands. The probability is 4/2,598,960, which is 1 in 649,740.
- Four of a kind. Choose the rank (13 ways), take all four cards of it, then choose any one of the other 48 cards: 13 × 48 = 624 hands.
- Any flush including straight flushes. Choose a suit (4 ways), then choose 5 of its 13 cards: C(13, 5) = 1,287. So 4 × 1,287 = 5,148 hands. This count includes the 40 straight flushes (including the royal flushes), which are normally ranked separately.
The rarity of the top hand also explains why the royal flush drives so much of the swings in video poker.
Choosing the right tool
A quick test helps decide which method to use.
- Does the order of selection matter to the question? If yes, use a permutation.
- If only the group matters, use a combination.
- If the selection comes from separate groups, count each group’s choices and multiply.
- If you are counting “at least one,” consider the complement: count the opposite and subtract from the total.
A common mistake is using permutations when the game only cares about the set of cards, which inflates the denominator and the numerator inconsistently. Another is forgetting that cards are removed once dealt, which links later draws to earlier ones.
A note on hand probabilities and payouts
Knowing how rare a hand is does not tell you whether a bet is worth its price. Rarity must be compared with the payout, and that comparison is the job of expected value. For the cards in a casino setting, the counts here feed into the pay table of a game, and the house edge is determined by how closely the pay table matches true frequencies.
Counting skills are useful for understanding games. They do not guarantee any result, and every outcome remains uncertain.