Combinations, permutations and arrangements
Count how many ways you can choose or order items, with or without repetition.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- Count how many ways you can choose or order items, with or without repetition.
- Formula used
C(n,k) = n! / (k!·(n−k)!)- Example
- Lottery 6 from 49: order does not matter and there is no repetition.
Combinations and permutations calculator
The difference between the five counts comes down to two questions: does order matter, and can an item repeat?
Once you have answered those, the right formula is determined.
How it works
If order does not matter and nothing repeats, they are combinations. If order matters, they are permutations of k from n. When you take all the items and order matters, they are full permutations.
Formula
C(n,k) = n! / (k!·(n−k)!)
CR(n,k) = (n+k−1)! / (k!·(n−1)!)
P(n,k) = n! / (n−k)!
PR(n,k) = n^k
P(n) = n!
Worked example
Lottery 6 from 49: order does not matter and there is no repetition.
C(49,6) = 49! / (6!·43!) = 13,983,816
That is the chance of a jackpot: 1 in nearly 14 million.
Explanation
Order decides which formula to use
If order matters, they are permutations: the top three places in a race are not the same depending on who finishes first. If it does not, they are combinations: six lottery numbers are worth the same in any order. Choosing wrongly between them is the commonest error, and gives results differing by orders of magnitude.
Why the lottery is so improbable
Choosing six numbers from forty-nine gives almost fourteen million possible combinations. That number does not come from nowhere: it is the direct result of the combinations formula, and it explains why buying twice as many tickets barely moves the real probability.
The birthday paradox
In a group of twenty-three people, the probability that two share a birthday exceeds fifty per cent. It seems impossible with three hundred and sixty-five days available, but what grows quickly is not the number of people but the number of possible pairs between them.
With or without repetition
If an element can repeat, as in a password where the same character can appear several times, the calculation changes completely. It is the difference between a padlock combination and a hand of cards.
Frequently asked questions
When do I use permutations instead of combinations?
When order changes the outcome. A podium of three from ten athletes is a permutation; picking three finalists is a combination.
What are combinations with repetition?
Cases where the same item can be chosen more than once, such as ordering three scoops from five ice cream flavours.
When do I use combinations and when permutations?
Combinations when order does not matter, as in a lottery. Permutations when it does, as on a podium or in a password.
How many combinations in a 6 from 49 lottery?
Almost fourteen million. That is why doubling your tickets barely changes the probability of winning.
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