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Basics
Combinations: how many ways?
A lottery ticket is a combination: a set of numbers where order does not matter. Counting combinations is the first step to any odds or probability question about a game.
Order doesn't matter
Drawing 4, 17, 23, 31, 38, 42 is the same ticket as 42, 38, 31, 23, 17, 4. The number of distinct tickets is therefore C(pool, k) — not pool × (pool−1) × …, which would count each ticket many times.
The numbers get big fast
C(49,6) is 13,983,816. C(50,5) × C(12,2) — EuroMillions — is 139,838,160. C(69,5) × 26 — Powerball — is 292,201,338. The difference between games is not a rounding error; it is the entire story of jackpot odds.