Counting is half of probability
When outcomes are equally likely, a probability is a count of favourable cases divided by a count of all cases. So many questions come down to counting well. How many possible poker-style hands exist? How many ways can a team be chosen? Combinations and permutations are the two core tools.
The multiplication principle
If one choice can be made in a ways and a second, independent choice in b ways, the pair can be made in a × b ways. Rolling two dice gives 6 × 6 = 36 ordered outcomes. Three dice give 6 × 6 × 6 = 216.
Everything else builds on this principle.
Permutations: order matters
A permutation is an ordered arrangement. Suppose you must pick 3 cards from a set of 5, and the order of picking matters (first, second, third).
- 5 choices for the first card.
- 4 remaining for the second.
- 3 remaining for the third.
- Total: 5 × 4 × 3 = 60 ordered arrangements.
The general formula for arranging r items out of n is n! / (n − r)!, where n! (n factorial) is n × (n − 1) × … × 1. Here, 5! / 2! = 120 / 2 = 60, which matches. If you arrange all n items, the count is just n!. Four distinct items can be ordered in 4! = 24 ways. A full 52-card deck has 52! orderings, a number so huge that a well-shuffled deck is, for practical purposes, almost certainly in an order that has never occurred before. Factorials get more attention under factorials.
Combinations: order does not matter
A combination is an unordered selection. If you only care which 3 cards you hold, not the order they arrived in, the 60 ordered arrangements overcount. Each unordered set of 3 appears in 3! = 6 different orders, so divide:
60 / 6 = 10 combinations.
The general formula for choosing r items from n is:
C(n, r) = n! / ( r! × (n − r)! )
Check: C(5, 3) = 120 / (6 × 2) = 10. ✓.
Other quick examples:
- Choosing 3 party members from 8 candidates: C(8, 3) = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56.
- Choosing 2 cards from 52: C(52, 2) = (52 × 51) / 2 = 1,326.
The decision rule is the question “does swapping the order give a different result?” If yes, use permutations. If no, use combinations. The difference between them is the focus of order matters.
Hand counts
Poker-style hands are unordered sets of cards, so combinations apply. A five-card hand from a standard deck:
C(52, 5) = (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1) = 311,875,200 / 120 = 2,598,960.
That is the well-known total of 2,598,960 five-card hands. Reasoning about particular hands then becomes a matter of counting how many of those 2,598,960 qualify, a topic that has its own treatment under five card hands.
Two-card starting hands: C(52, 2) = 1,326. How many of them are a pair of aces? There are 4 aces and we choose 2: C(4, 2) = 6. So the chance of being dealt two aces is 6 / 1,326 = 1/221, about 0.45%. This matches the sequential calculation 4/52 × 3/51 = 1/221, which is a good check that the methods agree.
Draw outcomes
The same counting handles questions about draws. What is the chance that a 3-card draw from a standard deck contains exactly 2 hearts?
- Ways to choose 2 hearts from 13: C(13, 2) = 78.
- Ways to choose 1 non-heart from 39: C(39, 1) = 39.
- Favourable draws: 78 × 39 = 3,042.
- All 3-card draws: C(52, 3) = (52 × 51 × 50) / 6 = 132,600 / 6 = 22,100.
- Probability: 3,042 / 22,100, about 0.1376, or 13.8%.
The pattern of “choose the favourable ones, choose the rest, multiply, divide by all combinations” is the engine behind the hypergeometric distribution, which describes draws without replacement.
Mixed cases: a lock is not a combination
A “combination lock” is a misnomer in this vocabulary. Since the order of digits matters, a 3-digit code from digits 0 to 9, with repeats allowed, has 10 × 10 × 10 = 1,000 possibilities. These are ordered selections, so they are arrangements, not combinations in the mathematical sense. This mismatch between everyday and mathematical language is explored under combination locks.
Common mistakes
- Using ordered counts for unordered questions. This overcounts by a factor equal to the number of orderings.
- Forgetting that repeats change the formula. Dice rolls allow repeated values; a deck draw without replacement does not.
- Mixing counting methods in numerator and denominator. If the denominator counts ordered draws, the numerator must too.
- Hand-wavy overlap. When cases overlap, add and subtract carefully using the inclusion-exclusion principle.
Checking your work
Whenever possible, calculate the same probability two ways, as above with the pair of aces. If they agree, confidence is high. Small cases, like choosing 2 of 4, can also be listed in full to confirm the formula.
A note on use
These tools describe fair random draws and explain why certain hands are rarer than others. They do not alter the expected return of a game, which for casino games favours the operator. Outcomes remain uncertain, and anyone struggling with gambling should contact a local support service.