Combinations, variations and permutations
Calculate how many ways you can choose k elements out of n. With order taken into account or without it.
How to use this page
- Fill in the fields in the form above — the result recomputes at once, with no page reload.
- Under the result you will find the interpretation scale, if the calculator has one: it says which range your value fell into.
- The formula we use is shown lower on this page — you can check the arithmetic yourself.
- To compute the same for another game or variant, change it in the select field — prices and pools update automatically.
What this page does not do: It does not point to numbers to play and does not predict a draw — every draw is independent of the previous ones.
Further reading
Combinatorics answers one question: in how many ways. The single thing that separates combinations from variations is the role of order. When you draw six numbers in Lotto, order plays no role — the set 3, 11, 24, 31, 38, 49 is the same as 49, 38, 31, 24, 11, 3. That's a combination. When placing people on a podium, order decides everything — that's a variation or a permutation.
An example from Lotto. The pool is 49 numbers, you pick six. The number of combinations without repetition is 13,983,816. That means the probability of hitting the six-number jackpot with one bet is roughly one in almost fourteen million. For Mini Lotto, where you choose five out of forty-two, there are only 850,668 combinations — which is sixteen times fewer than in Lotto. Exactly the same calculation applies to lotto systems: a full system of eight numbers is combinations of eight taken six at a time, meaning 28 bets.
Results grow faster than intuition suggests. A permutation of ten elements is already over three million arrangements, and of twenty — over two and a half quintillion. We've limited n to 170 — beyond that the result no longer fits in a standard numeric type. In modes without repetition, k cannot be greater than n, because you can't choose seven different numbers out of six.
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Frequently asked questions
How many combinations are there of six numbers out of forty-nine?
Exactly 13,983,816. That's how many different tickets you'd need to fill in to be certain of the Lotto jackpot.
What's the difference between a combination and a variation?
A combination ignores order, a variation does not. Out of 5 elements taken 3 at a time there are 10 combinations, but as many as 60 variations — six times more, because three elements can be arranged in 6 ways.
How many bets does a full system of 8 numbers have in Lotto?
28 bets, since that's the number of combinations of 6 out of 8. At a price of 3 zl per bet, such a system costs 84 zl.
Why doesn't the calculator accept n greater than 170?
Because 171! exceeds the maximum value of a floating-point number and the result would be infinity. Larger calculations require arbitrary-precision arithmetic.
See also
Arithmetic Mean
Enter numbers separated by commas and calculate the arithmetic mean online. The calculator will give you the s
Geometric mean
Calculate the geometric mean of any number of values. The n-th root of the product. Useful for rates of return
Hypergeometric Distribution Calculator
Probability of k successes when drawing without replacement: P(X = k), P(X ≤ k), P(X ≥ k), the full table, mean and variance — with urn, card and lottery examples.