Why counting comes first
Many chance questions look hard only because the number of possible outcomes is large. When every outcome is equally likely, the probability of an event is the number of outcomes you are interested in divided by the total number of outcomes. So the real work is counting, and combinatorics is the set of techniques that does the counting without listing everything one by one.
This page is a tour of the main tools. Each one has its own deeper treatment elsewhere in this cluster, so the aim here is to show what each tool is for and when to reach for it.
One caution before starting: counting tells you how likely something is, not how to make it happen. In any game with a built-in disadvantage for the player, such as most casino games, the maths describes why money is lost over time. It does not offer a way around that.
Tool 1: multiply when choices come in stages
Suppose a game lets you pick one of 3 characters and one of 4 starting items. Each character can go with each item, so there are 3 × 4 = 12 combinations. If you also pick one of 2 colours, the total is 3 × 4 × 2 = 24.
This is the multiplication principle: when a process has several independent stages, multiply the number of options at each stage. Rolling two six-sided dice gives 6 × 6 = 36 outcomes, which is the starting point for most dice probability questions.
Tool 2: factorials count orderings
How many ways can 4 people line up? There are 4 choices for the first place, 3 for the second, 2 for the third and 1 for the last: 4 × 3 × 2 × 1 = 24. That product is written 4! and called “four factorial”. Factorials grow fast: 10! is already 3,628,800. They show up inside almost every formula below.
Tool 3: permutations when order matters
A permutation is an arrangement in which the order counts. If 5 players race and only the first three places are recorded, the number of possible podiums is 5 × 4 × 3 = 60. First place has 5 candidates, second has 4 left, third has 3 left.
Order matters whenever swapping two items gives a different result: finishing positions, passwords, seat assignments, the sequence in which cards are dealt.
Tool 4: combinations when order does not matter
Now suppose you only want to know which 3 of the 5 players are selected for a team, with no ranking. Every group of 3 corresponds to 3! = 6 different podium orders, so the 60 ordered results contain each group 6 times. Divide: 60 ÷ 6 = 10 possible teams.
The general idea is to count ordered selections and then divide by the number of ways to reorder each selection. A hand of 5 cards from a standard deck has 52 × 51 × 50 × 49 × 48 = 311,875,200 ordered deals, and each hand can be arranged in 5! = 120 ways, so there are 311,875,200 ÷ 120 = 2,598,960 different hands. The same reasoning underlies lottery odds, where you pick a set of numbers and the order they are drawn in does not matter. For a format that draws 6 numbers from 49, the count of possible sets is 13,983,816.
Tool 5: Pascal’s triangle and binomial coefficients
The numbers produced by “choose k from n” form a pattern known as Pascal’s triangle. Each entry is the sum of the two above it, and the entries tell you how many ways there are to get exactly k successes in n attempts. For five coin flips, the row reads 1, 5, 10, 10, 5, 1, so there are 10 ways to see exactly two heads out of 32 equally likely sequences.
Tool 6: count the opposite
Questions with “at least one” in them are often easier from the other side. The chance of at least one six in four rolls of a die equals 1 minus the chance of no sixes at all. With no sixes the count is 5 × 5 × 5 × 5 = 625 out of 6 × 6 × 6 × 6 = 1,296 sequences. So the chance of at least one six is 1 − 625/1296 = 671/1296, which is about 51.8%.
Tool 7: inclusion-exclusion for overlapping groups
If you add up two groups that overlap, the overlap gets counted twice. Take a standard deck: there are 13 hearts and 4 kings, but the king of hearts is in both groups. The number of cards that are a heart or a king is 13 + 4 − 1 = 16 out of 52. Subtracting the overlap is the first step of inclusion-exclusion.
Tool 8: trees for multi-stage problems
When the second stage depends on what happened in the first, a probability tree keeps the bookkeeping straight. Each branch carries a probability, and you multiply along a branch to get the chance of that full path. Trees are also the natural way to see why conditional probability changes the numbers when cards are removed from a deck.
Tool 9: stars and bars for sharing identical items
If you want to know how many ways 5 identical tokens can be split among 3 players, the items are indistinguishable and only the amounts matter. The stars and bars picture gives C(7, 2) = 21 ways. It is a good tool when a problem is about quantities rather than about who holds which object.
Choosing the right tool
A short checklist helps:
- Are there stages of independent choices? Multiply.
- Are you selecting items and does the order of the selection matter? If yes, use a permutation count. If no, use a combination count.
- Does the question say “at least”? Try the opposite event first.
- Do the groups you are adding overlap? Subtract the overlap.
- Are the stages linked, with later chances depending on earlier results? Draw a tree.
- Are you splitting identical items? Think stars and bars.
Equal likelihood is the hidden assumption behind all of this. A count of favourable outcomes divided by a count of total outcomes only gives a probability when each outcome has the same chance. If a loaded die or an uneven prize table is involved, counting alone is not enough and you need to weight the outcomes. Once the probabilities are in place, ideas such as expected value let you compare uncertain options, and in gambling contexts they typically show that the house edge works against the player over many plays.
Hand checks matter too. Where a count is large, it is worth confirming it with a smaller version of the problem, or with a Monte Carlo simulation, before trusting it.
A note on gambling
The examples in this cluster use dice, coins and cards because they are easy to count, not because they suggest ways to profit. Past results never change the odds of the next independent event, and games with a house edge lose money over time for the typical player. Gambling can be harmful, and anyone who is struggling should contact a local support service.