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 only difference between a combination and a variation is whether order matters. 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 you line up six people in a queue, order changes everything — that's a variation or a permutation.
An example from Lotto. You choose six numbers out of forty-nine. The number of combinations without repetition is 13,983,816. The chance of hitting six numbers with a single ticket is about 1 in 14 million. In Mini Lotto, picking five from 42 numbers, there are 850,668 possibilities — which is sixteen times fewer than in Lotto. The same math governs shortened and full systems: a full system of eight numbers is combinations of eight taken six at a time, meaning 28 bets.
Watch out for the order of magnitude. Ten elements can be arranged in 3,628,800 ways, twenty elements in over 2.4 quintillion ways. We've limited n to 170 — beyond that the result no longer fits in a standard numeric type. In variants without repetition, the number of elements chosen cannot exceed the whole pool, because you can't choose seven different numbers out of six.
Read more
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.