Why counting comes first
Most basic probability questions reduce to one fraction: the number of outcomes you care about divided by the number of outcomes that are possible, when those outcomes are equally likely. The arithmetic of that fraction is easy. The hard part is counting both numbers without missing cases or counting the same case twice.
This page collects the four tools that handle most of those counting problems: the multiplication principle, factorials, permutations and combinations. Each is shown with a small game example whose numbers you can verify by hand. The ideas sit underneath broader probability theory, and they are the reason a lot of “surprisingly large” numbers in games turn out to be ordinary arithmetic.
The multiplication principle
If one choice can be made in a ways, and for each of those a second choice can be made in b ways, the two choices together can be made in a × b ways. Extend this to as many stages as you need by multiplying them all.
Rolling two six-sided dice is the standard example. The first die has 6 faces and the second has 6, so there are 6 × 6 = 36 ordered outcomes. Rolling a die three times gives 6 × 6 × 6 = 216.
The same idea works when the number of options changes from stage to stage. Suppose a board game lets you pick one of 4 characters, then one of 3 starting items, then one of 2 starting locations. That is 4 × 3 × 2 = 24 possible setups. The key test is whether the stages are really independent in the sense of counting: the number of options at a later stage must not depend on which option you picked earlier, or you must account for how it does.
Counting every outcome like this is what makes dice sums easy to reason about, since any total is a count of some of the 36 outcomes divided by 36. That is the whole story behind two dice totals.
Factorials: counting orderings
Suppose you have n different items and want to line them all up. There are n choices for the first position, n − 1 for the second, n − 2 for the third, and so on down to 1. By the multiplication principle the total is
n × (n − 1) × (n − 2) × … × 1
which is written n! and read “n factorial”.
Some small values: 3! = 6, 4! = 24, 5! = 120, 10! = 3,628,800. By convention 0! = 1, which keeps later formulas tidy.
Factorials grow very quickly. A standard 52-card deck can be arranged in 52! ways, a number of about 8.07 × 10^67, which is a 68-digit number. That is why a properly shuffled deck is almost certainly in an order that has never occurred before. The details are in the article on factorials, and there is a neighbouring question about shuffled orderings in which nothing lands in its original place, covered under derangements.
Permutations: choosing some items where order matters
Often you do not arrange all n items, only k of them. Picking and ordering k items from n distinct items can be done in
n × (n − 1) × … × (n − k + 1)
ways, which equals n! ÷ (n − k)!. This is called a permutation count.
Example: eight runners are in a race and the first three places are gold, silver and bronze. The number of possible podiums is 8 × 7 × 6 = 336. Winning order is part of the outcome, so each different order is a different result.
A second example is a four-digit code where no digit may repeat. With 10 digits available, there are 10 × 9 × 8 × 7 = 5,040 codes. When digits may repeat, the multiplication principle gives 10 × 10 × 10 × 10 = 10,000, which is the kind of growth explored with combination locks.
Combinations: choosing some items where order does not matter
Sometimes only the group matters. Choosing 3 items from 8 as a set, with no ranking, is a combination. Each group of 3 can be ordered in 3! = 6 ways, and the permutation count 336 counts every ordering separately, so
336 ÷ 6 = 56
different groups of 3. In general, the number of combinations of k items from n is n! ÷ (k! × (n − k)!), often written “n choose k”.
Another example: a party of 4 adventurers needs 2 of them to scout ahead. The number of possible scouting pairs is 4 × 3 ÷ 2 = 6. If instead one is the leader and one is the follower, the roles make the order matter, and the count is 4 × 3 = 12.
Whether roles make order matter is the heart of the difference between arrangements and selections. The distinction is developed with more examples in the discussion of when order matters, and the broader family of combinations and permutations is a common source of mistakes in games with hands of cards.
A worked card example
How many different five-card hands can be dealt from a standard 52-card deck, ignoring the order cards arrive in? The ordered count is 52 × 51 × 50 × 49 × 48 = 311,875,200. Each hand can arrive in 5! = 120 orders, so the number of hands is 311,875,200 ÷ 120 = 2,598,960.
That figure is the denominator of every poker hand odds calculation that deals five cards from a full deck. The number of hands of each type is then a separate counting job, which is worked through in the article on five card hands.
Beyond the four basics
Several other techniques build on these. When events overlap, adding their counts double counts the shared part, and the fix is the inclusion-exclusion principle. When you want to split identical items among people or boxes, stars and bars is the standard method. The coefficients you get from combinations also appear in Pascals triangle, which is a convenient way to see small cases. And sometimes counting alone proves something is guaranteed, which is what the pigeonhole principle does.
Common slips to watch for
- Mixing up order. Decide before counting whether swapping two chosen items gives a new outcome or the same one.
- Forgetting restrictions. If some choices are not allowed after earlier choices, the numbers in later stages shrink.
- Counting outcomes that are not equally likely as if they were. Dividing favourable by total only works when each counted outcome has the same chance.
- Double counting. If one outcome can be described in two ways, make sure it enters the count once.
A note on games of chance
Counting tells you how many outcomes are possible and, if they are equally likely, how often a particular kind should appear over many trials. It does not predict the next result, and it does not remove the built-in disadvantage in casino games, where the house edge means players lose on average over time. Gambling can be harmful; anyone who finds it hard to stop should consider contacting a local support service.