How lottery odds are actually calculated
Lottery jackpot odds are counted, not guessed. A play must match a fixed set of numbers in any order (a combination), so the count uses C(n,k) = n!/(k!(n-k)!) for the main numbers, then multiplies by the bonus-ball pool. For Powerball that gives C(69,5) x 26 = 292,201,338, exactly matching the published 1 in 292,201,338 figure.
Why show the arithmetic at all
Every odds figure on this site is printed by the game itself, and every odds figure can also be derived from scratch with nothing more than counting. This page does the counting, so the numbers on the other pages do not have to be taken on trust.
What a combination is, and why order does not matter
A lottery play is a combination: an unordered selection. If a drawing pulls 7, 19, 33, 41, 58, it does not matter whether your ticket lists them in that order or in the order 58, 7, 41, 19, 33 — either way, it is the same winning ticket. That is different from a permutation, where order counts.
Because order does not matter, the count of possible plays is smaller than it would be if it did. Counting ordered arrangements of 5 numbers chosen from 69 gives 69 x 68 x 67 x 66 x 65, but that count treats "7-19-33-41-58" and "58-41-33-19-7" as two different outcomes, when a lottery ticket treats them as one. Every group of 5 numbers can be arranged in 5 x 4 x 3 x 2 x 1 = 120 different orders, so the ordered count is exactly 120 times too big. Dividing by 120 corrects for that, and that division is the entire reason the formula has a k! in it.
The formula
Written formally, the number of ways to choose k items from a set of n, with order not mattering, is:
C(n,k) = n! / (k! × (n-k)!)
Read "n choose k." The n! on top counts every ordered arrangement of all n items; dividing by (n-k)! removes the part of that arrangement beyond the k items actually chosen, leaving every ordered arrangement of just the k chosen items; dividing by k! then removes the ordering within those k items, which is the step from the previous section.
A jackpot requires two independent things to both happen: the k main numbers drawn must be exactly the k the player picked (in any order), and the separately drawn bonus number must match too. Two independent requirements combine by multiplying their counts, so:
The number of ways to choose k items from a set of n, order not mattering, is C(n,k) = n! / (k! × (n-k)!), and jackpot odds = 1 in [C(main pool, main picks) × bonus pool]
Powerball, worked through
Powerball draws 5 white balls from 69 and 1 red Powerball from 26. Plugging those into the formula:
C(69,5) x 26
- 1
C(69,5) = (69*68*67*66*65) / (5*4*3*2*1) = 1,348,621,560 / 120 = 11,238,513 - 2
11,238,513 x 26 (Powerball choices) = 292,201,338
Odds of one play winning the jackpot1 in 292,201,338
That figure matches the one the game publishes, which is the point of doing it here rather than taking it on trust.
C(69,5) counts every possible group of 5 white numbers, ignoring order: multiply 69 by the four numbers below it (68, 67, 66, 65), then divide by 120 to undo the ordering, and the result is 11,238,513 distinct white-ball groups. Each of those groups can be paired with any of the 26 possible Powerballs, and only one specific pairing is the winning one, so multiplying by 26 gives the total number of equally likely plays: 292,201,338. A single ticket is a 1-in-that-many chance, which is exactly the figure Powerball (Multi-State Lottery Association), 2026 publishes.
Mega Millions, the same method
Mega Millions draws 5 white balls from 70 and 1 gold Mega Ball from 24 (the current, post-April-2025 matrix). The same method applies, just with different numbers:
| Step | Calculation | Result |
|---|---|---|
| Combinations of 5 from 70 | (70×69×68×67×66) / (5×4×3×2×1) | 12,103,014 |
| Multiply by the Mega Ball pool | 12,103,014 × 24 | 290,472,336 |
That result, 290,472,336, is exactly the figure Mega Millions Consortium, 2026 publishes for the current Mega Millions matrix. The only difference from the Powerball calculation is the two pool sizes: one more white number to choose from (70 instead of 69), and two fewer bonus-ball choices (24 instead of 26).
A lower tier, derived the same way: matching 4 plus the Powerball
The same counting method works for every prize tier, not just the jackpot. Here is the derivation for one of the more valuable lower tiers: matching exactly 4 of the 5 white numbers, plus the Powerball.
To match exactly 4 white numbers and no more, a winning ticket needs 4 correct white numbers from the 5 drawn, and its 5th white number has to be wrong — chosen from the 64 white numbers that were not drawn (69 total minus the 5 that were drawn). Counting the ways to do that:
| Step | Calculation | Result |
|---|---|---|
| Ways to pick which 4 of the 5 drawn white balls you matched | C(5,4) | 5 |
| Ways to pick your one wrong white ball, from the 64 not drawn | C(64,1) | 64 |
| The Powerball must also match (fixed, not a further choice) | × 1 | × 1 |
| Winning combinations for this tier | 5 × 64 × 1 | 320 |
320 tickets out of 292,201,338 possible plays win this tier, so the odds are 292,201,338 / 320 = 913,129.18, which rounds to the published 1 in 913,129.18. The same logic, with Mega Millions' 65 unmatched white numbers instead of 64, gives that game's 4-plus-Mega-Ball tier: C(5,4) × C(65,1) = 5 × 65 = 325 winning combinations out of 290,472,336, which works out to 1 in 893,761 — Mega Millions Consortium, 2026 confirms both figures.
Why this is worth checking rather than trusting
The point of doing this arithmetic in public is that anyone can redo it. Every result above matches the number the game itself publishes, which is the actual test of whether the method is right — not that the number looks plausible, but that it lands exactly on the figure printed on the official prize chart.
This also explains why prize odds sit where they do relative to each other. Requiring an exact match on more numbers, or on a bigger pool, shrinks the count of winning combinations fast, because each additional requirement is another multiplication by a small number rather than an addition. That is why the jump from "match 4 plus the bonus ball" to "match 5 plus the bonus ball" is a jump of hundreds of times in odds, not a small step.
How the odds are calculated: FAQ
What formula gives lottery jackpot odds?
C(n,k) = n!/(k!(n-k)!) for the main-number pool, multiplied by the size of the separate bonus-ball pool, since a ticket must match both independently.
Why do you divide by k! in the combination formula?
Because a lottery ticket does not care what order its numbers were drawn in. Dividing by k! (the number of ways to arrange k items) removes the extra count created by treating different orderings of the same numbers as different outcomes.
How is C(69,5) calculated for Powerball?
(69x68x67x66x65) divided by (5x4x3x2x1), which equals 11,238,513. Multiplying by the 26 possible Powerballs gives 292,201,338, the published jackpot odds.
How is the Mega Millions jackpot figure of 290,472,336 calculated?
C(70,5) x 24: choosing 5 white numbers from 70 gives 12,103,014 combinations, multiplied by the 24 possible Mega Balls in the current matrix.
How do you calculate the odds of a lower prize tier, like matching 4 numbers plus the bonus ball?
Count the ways to get exactly that match: choose which 4 of the 5 drawn white numbers you matched (C(5,4) = 5), choose your one wrong white number from the ones not drawn (C(64,1) = 64 for Powerball), then divide the total number of plays by that count (320) to get the odds.
Does this arithmetic match what the games publish?
Yes. Every figure calculated on this page, for both games and multiple tiers, matches the odds printed on the official Powerball and Mega Millions prize charts exactly or to the nearest published rounding.
How the odds are calculated: common questions
What formula gives lottery jackpot odds?
C(n,k) = n!/(k!(n-k)!) for the main-number pool, multiplied by the size of the separate bonus-ball pool, since a ticket must match both independently.
Why do you divide by k! in the combination formula?
Because a lottery ticket does not care what order its numbers were drawn in. Dividing by k! (the number of ways to arrange k items) removes the extra count created by treating different orderings of the same numbers as different outcomes.
How is C(69,5) calculated for Powerball?
(69x68x67x66x65) divided by (5x4x3x2x1), which equals 11,238,513. Multiplying by the 26 possible Powerballs gives 292,201,338, the published jackpot odds.
How is the Mega Millions jackpot figure of 290,472,336 calculated?
C(70,5) x 24: choosing 5 white numbers from 70 gives 12,103,014 combinations, multiplied by the 24 possible Mega Balls in the current matrix.
How do you calculate the odds of a lower prize tier, like matching 4 numbers plus the bonus ball?
Count the ways to get exactly that match: choose which 4 of the 5 drawn white numbers you matched (C(5,4) = 5), choose your one wrong white number from the ones not drawn (C(64,1) = 64 for Powerball), then divide the total number of plays by that count (320) to get the odds.
Does this arithmetic match what the games publish?
Yes. Every figure calculated on this page, for both games and multiple tiers, matches the odds printed on the official Powerball and Mega Millions prize charts exactly or to the nearest published rounding.
Keep reading
Powerball odds
The full tier-by-tier table these calculations feed, with Power Play and Double Play explained.
Read →Mega Millions odds
The full tier-by-tier table for Mega Millions, including the April 2025 format change.
Read →Odds in perspective
The jackpot number next to things people can actually picture, with the arithmetic shown.
Read →The odds do not move, so check them yourself
Powerball is 1 in 292,201,338 and Mega Millions is 1 in 290,472,336 for one play. Every figure on this site is dated, sourced and shown with its arithmetic, and we sell nothing that could change any of it.