Combinations, variations and permutations
How many different sets can you build from n elements, choosing k. Four variants: with and without repetition, with and without order.
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. 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.
Results grow faster than intuition suggests. Ten elements can be arranged in 3,628,800 ways, twenty elements in over 2.4 quintillion ways. The calculator accepts n up to 170, because above that limit the factorial exceeds the range of floating-point numbers. In modes without repetition, k cannot be greater than n, because you can't choose seven different numbers out of six.
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
Nth Root Calculator
Compute the square root, cube root or a root of any degree. The calculator handles negative numbers for odd de