Combinatorics explained for betting

In one sentence

Combinatorics is the maths of counting: how many ways a set of selections can be combined or ordered.

How it works

Many bets are really bundles of smaller bets. A Lucky 15 on four football results is fifteen bets: four singles, six doubles, four trebles and one fourfold. Knowing how the count works tells you what you are staking and what you can win.

Two kinds of counting matter. Combinations ignore order: picking which 2 of 4 teams go in a double. Permutations care about order: naming the champions and runners-up in the right sequence.

Once you can count the outcomes, you can work out probabilities and returns. It is simple maths, but mistakes here lead to over-staking and misreading returns.

The maths

\binom{n}{k} = \frac{n!}{k!\,(n-k)!}

P(n, k) = \frac{n!}{(n-k)!}

n: the number of selections or teams.
k: how many you choose for each bet or position.
n! (n factorial): n × (n − 1) × ... × 1.
The first formula counts combinations, where order does not matter.
The second counts permutations, where order matters.

In plain English: the first tells you how many different groups you can pick, the second how many different finishing orders you can name.

Worked betting example

A Lucky 15 on four home wins at £1 per line.

Singles: 4 choose 1 = 4. Doubles: 4 choose 2 = 6. Trebles: 4 choose 3 = 4. Fourfold: 4 choose 4 = 1.
Total lines = 4 + 6 + 4 + 1 = 15, so total stake = £15.

The odds are 3.00, 4.00, 2.50 and 5.00. Only the 3.00 and 2.50 teams win.

Winning singles return £3.00 + £2.50 = £5.50.
The one double with both winners returns 3.00 × 2.50 = £7.50.
Every treble and the fourfold include a loser, so return nothing.
Total return = £5.50 + £7.50 = £13.00, a loss of £2 on £15 staked despite two of four winners.

Ordering in a 20-team league, and in a scoreline:

Champions and runners-up in order: 20 × 19 = 380 possible results.
Top three in order: 20 × 19 × 18 = 6,840.
Top three in any order: 20 choose 3 = 1,140.
A 3-2 final score: the two away goals can fall anywhere among the five, so 5 choose 2 = 10 possible goal orders.

So even if every team were equally likely, a single top-three-in-order line would win 1 time in 6,840.

Where it's good

Understanding exactly what a full-cover bet (Yankee, Lucky 15, Heinz) costs and pays.
Pricing finishing-order and goal-order outcomes from win probabilities.
Checking that a multiple's odds match its legs.
Counting outcomes for correct score, scorecast and exact-order markets.
Planning dutching and covering bets across many selections.

Limitations and pitfalls

Counting outcomes is not the same as pricing them. Teams are rarely equally likely, so each combination needs its own probability.
Multiples assume the legs are independent. Linked legs make the true probability differ from the product.
Full-cover bets compound the bookmaker's margin on every line, which is why they are profitable for bookmakers.
The number of lines grows fast. Six selections in a full-cover bet is 57 lines; stakes balloon quietly.
Bonuses on full-cover bets can look generous but rarely offset the compounded margin.

How to build it

Python's math.comb, math.perm and itertools.combinations are enough to count and list every line.
Data: odds per selection, and win probabilities if you want the EV of each line.
Practical tip: generate every line with itertools and compute its return and probability in a table. It exposes where the money in a full-cover bet really comes from.

Independence and correlation: when multiplying leg probabilities is valid.
Binomial: the chance of k winners from n selections.
Plackett-Luce and Harville: turning win chances into forecast and tricast probabilities.
Expected value: judging each line once you have counted it.