Combinations vs Permutations with Cards
Combinations and permutations compared on a 52-card deck, and why a poker hand is not an ordered deal.
People mix up two counting questions that look alike. How many ways can five cards be dealt in a row, and how many distinct five-card poker hands exist? The first question cares about order. The second does not. The names are permutations and combinations. Getting them backward will inflate a poker probability by a factor of 120, which is enough to invent a completely different game.
A three-card picture
Take three cards labeled A, B, and C, and choose two. If order matters, the list is AB, BA, AC, CA, BC, and CB. Six results. You can also compute 3 times 2, because the first card has three choices and the second has two left. If order does not matter, AB and BA are the same hand. The list shrinks to AB, AC, and BC. Three results. The six ordered results were double-counting each hand, so you divide by 2.
That division is the whole difference. Permutations count sequences. Combinations count groups. A dealt poker hand is a group. The moment the dealer slides the cards does not create a new hand.
Five cards from a 52-card deck
Deal five cards in order from a 52-card deck and the first card has 52 options, then 51, 50, 49, and 48. Multiply those: 52 times 51 times 50 times 49 times 48 equals 311,875,200 ordered deals. Every unordered hand has been counted many times in that total, once for each way to arrange its five cards. The number of arrangements of five cards is 5 times 4 times 3 times 2 times 1, which is 120.
Divide 311,875,200 by 120 and you get 2,598,960. That is the number of distinct five-card hands. Quote the larger figure as the number of poker hands and every probability you build afterward is 120 times too small.
A royal flush, counted once
A royal flush is A, K, Q, J, 10 of one suit. There are four suits, so there are four such hands. The probability of being dealt one, from a shuffled deck, is 4 divided by 2,598,960, or 1 in 649,740. The four hands are combinations. If you had counted ordered deals, each royal would have contributed 120 sequences, and you would still divide those sequences by 120 before you finished. Skipping the division is the mistake, not the multiplication.
The same habit applies to any exact holding: a particular pair, a particular full house, a particular two-card start. Decide whether order matters, count, and only then divide if you counted sequences you consider identical. A short history of playing cards is background. It is not a substitute for the division.
Words that signal which count you need
How many hands usually means combinations. How many ways the cards can come off the deck in order means permutations. The chance the next card is an ace is neither a big combination nor a permutation. It is a single draw from the cards that remain. Mixing a remaining-card question with the 2,598,960 figure will produce a confident, wrong percentage.
Write the sample space in a sentence before you calculate. All unordered five-card hands, equally likely, is a different sentence from all ordered deals of five. Both can be valid. They are valid for different questions. A coin toss has no memory uses a sample space of four paths for two tosses. The discipline is identical. The objects are cards instead of tosses.
A check that catches the inflation
After you finish a count, see whether a smaller case agrees with it. Two cards from three cards should give three hands, not six, if order does not matter. If your method says six, you forgot to divide. Scale up only after the small case behaves. This will not tell you the house price of a game. It will stop you from announcing that a rare hand is 120 times rarer than it is.
The count is arithmetic, not an invitation to play. Age limits and local law still decide whether a card game is available to you.
Frequently asked questions
How many five-card poker hands are in a 52-card deck?
There are 2,598,960 unordered five-card hands. The larger count, 311,875,200, is the number of ordered deals of five cards.
Why divide by 120?
Because five cards can be arranged in 120 orders, and a poker hand treats those orders as the same hand.
Does this number include jokers?
No. It is a standard 52-card deck with no jokers. Add a joker and both counts change.