How this instrument works
A typical lottery jackpot requires matching every one of k numbers drawn from a pool of n possible numbers, with the order you pick them in not mattering — buying a ticket with 3, 17, 22 is exactly as good as 22, 17, 3. That's precisely what a combination counts: the number of ways to choose k items from n without regard to order, written C(n, k) and calculated as n! / (k!(n−k)!). The total number of distinct ticket combinations is exactly the number of ways the lottery could possibly draw its winning numbers, so your odds of matching the jackpot on a single ticket are exactly 1 in C(n, k).
Many real lotteries add a separate bonus number drawn from its own smaller pool — a Powerball, a Mega Ball — which must also be matched to win the full jackpot. Since the main-pool draw and the bonus-number draw are independent, the total number of combinations multiplies: total combinations = C(n, k) × bonus pool size, and the odds of matching everything become 1 in that larger product. This calculator handles both the no-bonus case (leave the bonus pool at 1) and the bonus-number case in the same formula.
These are combinatorial facts, not estimates — the odds calculated here are exact, deterministic consequences of the lottery's stated format, matching the officially published odds of real lotteries when the same pool sizes and pick counts are entered. What they can't tell you is anything about whether playing is a good idea; lottery odds are astronomically long by design, and this calculator is offered purely as an exercise in combinatorics, not a recommendation to play.
- Enter the size of the main number pool into Numbers in the main pool (n).
- Enter how many numbers must be matched into Numbers you pick / that are drawn (k).
- If the lottery includes a separate bonus number, enter that pool's size into Bonus-number pool size — leave it at 1 if there's no bonus number.
- Read Odds are 1 in this many combinations for the plain-language odds figure.
- Read Probability of winning (jackpot) for the same result expressed as a decimal probability.
Worked example — a classic 6/49 lottery
A classic 6/49-format lottery — pick 6 numbers from a pool of 49, no separate bonus number — is one of the most widely used lottery formats worldwide. Enter 49 into Numbers in the main pool (n), 6 into Numbers you pick / that are drawn (k), and leave Bonus-number pool size at 1.
C(49, 6) works out to 13,983,816 — a widely cited, well-documented combinatorial fact. Odds are 1 in this many combinations reads 13,983,816 exactly, and Probability of winning (jackpot) reads about 7.15×10⁻⁸ (roughly 1 chance in almost 14 million). Switching to a 5/69-plus-a-1/26-bonus format (the real US Powerball structure) instead gives C(69,5) = 11,238,513 main combinations, multiplied by the 26-number bonus pool for a total of 292,201,338 — matching the actual, widely published Powerball jackpot odds of 1 in 292,201,338.
Questions
Why doesn't the order of the numbers matter?
Because a lottery ticket wins by matching the set of drawn numbers, not the order they were drawn or written in — a ticket with 3, 17, 22 wins under exactly the same conditions as one with 22, 17, 3, since both represent the identical set of three numbers. That's precisely what a mathematical combination counts (as opposed to a permutation, which would count 3-17-22 and 22-17-3 as different outcomes), which is why lottery odds are calculated using the combinations formula, C(n, k), rather than a permutations formula.
How does adding a bonus number change the odds?
It multiplies the total number of combinations, because matching the main numbers and matching the separate bonus number are two independent requirements that both have to succeed. If the main pool gives C(n, k) possible combinations and the bonus pool has m numbers, the total number of distinct winning outcomes becomes C(n, k) × m, and the odds lengthen to 1 in that larger product — which is exactly why bonus-number lotteries like Powerball have such enormous odds compared to a plain single-pool lottery of similar size.
Are these odds exact, or an approximation?
They're exact — combinations are a precise counting result from combinatorics, not a statistical estimate or simulation. Given a lottery's stated pool size, pick count, and bonus-pool size, the number of possible combinations (and therefore the odds of any single ticket matching the winning draw) is a fixed mathematical fact, which is why this calculator's results match officially published lottery odds exactly when the same format numbers are entered.
Do my odds improve if I buy more tickets?
Buying additional tickets with different number combinations does improve your overall odds roughly in proportion to how many distinct tickets you hold, since each ticket is an independent chance at matching the draw — but the odds of any single ticket remain exactly 1 in the calculated combinations figure, and with odds this long (often in the tens or hundreds of millions to one), buying a modest number of extra tickets moves your overall chances only marginally.
Does a longer set of odds mean the lottery is unfair?
No — long odds are simply the direct mathematical consequence of the pool size and pick count a lottery chooses, not a sign of anything unfair about the draw itself. A larger pool or a higher pick count both increase the number of possible combinations sharply, which is exactly why lottery operators choose large pool sizes: it keeps jackpots rolling over and growing until someone eventually matches the draw.