How are lottery odds actually calculated?
Every major lottery's odds come from one branch of math called combinatorics — specifically, counting how many ways you can choose a set of numbers from a larger pool, where the order you pick them doesn't matter.
The formula is written C(n, k), read "n choose k": it counts how many different k-number combinations exist inside a pool of n numbers. For EuroMillions, you choose 5 main numbers from a pool of 50, and separately 2 Lucky Star numbers from a pool of 12. Multiply the two results together, because every possible set of 5 main numbers can be paired with every possible set of 2 stars:
C(50, 5) = 2,118,760 ways to pick the 5 main numbers.
C(12, 2) = 66 ways to pick the 2 Lucky Stars.
2,118,760 × 66 = 139,838,160 total possible combinations.
That's the entire calculation — no hidden variables, no machine-specific factors, no historical data. It only depends on two numbers: how large the pool is, and how many balls get drawn from it. That's also why odds vary so much between lotteries: SuperEnalotto's 6-from-90 format produces C(90,6) = 622,614,630 combinations — over 4x tougher odds than EuroMillions — purely because the pool is bigger and no bonus-number split exists to work in your favor the way EuroMillions' structure does.