One question decides everything
Both permutations and combinations count ways of selecting items from a group. The only difference is whether the order of selection matters. If the same items in a different order count as a different result, you want permutations. If they count as the same result, you want combinations.
Card games give clear examples of both. A poker hand is a set of cards: receiving the king of hearts first and the two of clubs second leaves you with the same hand as getting them in the opposite order. Order does not matter, so you count combinations. But a sequence of cards turned over one by one from a shuffled deck is ordered, and its sequence does matter, so you count permutations.
Permutations: order counts
The number of ordered selections of k items from n distinct items is n × (n − 1) × … down to k terms. For cards:
- Two cards dealt in order from 52: 52 × 51 = 2,652 sequences.
- Five cards dealt in order: 52 × 51 × 50 × 49 × 48 = 311,875,200 sequences.
Each factor is the number of cards still available at that step, because a card already dealt cannot be dealt again.
A smaller case is easier to see. With five distinct cards labelled A to E, how many ways can you draw three and keep track of order? 5 × 4 × 3 = 60. The sequences A-B-C and C-B-A are different entries in that list.
Combinations: order does not count
To count unordered selections, take the permutation count and divide by the number of ways to reorder the chosen items. Any group of k distinct cards can be arranged in k! = k × (k − 1) × … × 1 orders.
- A two-card hand: 2,652 ÷ 2 = 1,326.
- A five-card hand: 311,875,200 ÷ 120 = 2,598,960, since 5! = 120.
- Three cards from five: 60 ÷ 6 = 10.
You can verify the last one by listing: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE. That is ten groups, matching 60 ÷ 6.
The shortcut formula is n! ÷ (k! × (n − k)!), but you can always get there by dividing a permutation count by k!, which many people find easier to remember.
Two quick tests
- Would swapping two items give a different outcome? If yes, order matters.
- Am I counting a collection or a sequence? Hands, teams and groups of cards picked from a deck are collections. Passwords, finishing positions in a race, the order cards come off the top of the deck, and turn order are sequences.
When a problem is about a hand, you almost always want combinations, because the hand does not remember how it was assembled. When a problem is about shuffling or the exact order of a draw, permutations apply.
Why probability usually uses combinations for hands
Because every five-card hand is equally likely, the probability of a hand type is the number of hands of that type divided by 2,598,960. Both the top and the bottom should be unordered, or both ordered. Mixing them is a frequent error. For example, counting the favourable hands as ordered sequences while using 2,598,960 as the denominator gives an answer that is too high, by a factor of the number of orderings.
Here is a worked case. How many five-card hands contain all four aces? The fifth card can be any of the other 48 cards, so there are 48 hands. Probability: 48 ÷ 2,598,960, which is about 0.0000185, or roughly 1 in 54,145. Check: 2,598,960 ÷ 48 = 54,145. A quick note: this is just the chance of holding four aces, not four of a kind of any rank.
If instead you counted ordered sequences, the favourable count would be 48 × 120 = 5,760 and the denominator would be 311,875,200, which yields the same ratio, since 5,760 ÷ 311,875,200 = 1/54,145. Both routes agree as long as you are consistent. This is the main point: consistency matters more than the choice.
When order matters in card games
It is easy to think order never matters in cards, but several cases depend on it:
- Dealing from the top of the deck. If you want the chance that the top card is an ace and the next is a king, you are specifying a sequence: 4/52 × 4/51 = 16/2,652 = 4/663, about 0.60 percent. The reverse sequence, king then ace, has the same probability, but they are separate outcomes.
- Hold’em community cards. The flop is a set of three cards, while the turn and river are specified in order by timing. The final five-card board is usually treated as a set, but the order it arrived in may matter to your decisions along the way.
- Ordered game rules. In games where cards are played in sequence, such as laying cards down in a trick, the order of play changes who wins.
For the two-card starting hands in Hold’em, 1,326 combinations reduce to 169 distinct types when you ignore suit identity. That reduction is another form of “treating different results as the same” and is why starting hands are normally grouped.
Worked draw example
Suppose a deck has 40 cards, including 4 copies of a card you want. You draw an opening hand of 7 cards. The number of 7-card hands is “40 choose 7”. The number of hands containing no copy of your card is “36 choose 7”. The probability of at least one copy is 1 minus the ratio of those two. Working this through is a routine job for a calculator or spreadsheet, because the figures are large, but the setup is entirely a matter of choosing combinations, as these opening hands are unordered. The same structure lies behind the hypergeometric distribution.
Common slips to avoid
- Using the permutation formula for a hand, which overcounts by k!.
- Using the combination formula for a sequence, which undercounts by k!.
- Forgetting that items must be distinct. If some cards are identical in effect (say, copies of the same card in a deck), the counting of “different” results changes, and you must decide whether you are counting physical cards or distinct card types.
- Switching between ordered and unordered counts halfway through a problem.
If you remember only one thing, remember the conversion: permutations ÷ k! = combinations. Everything else is bookkeeping.