Permutations and Combinations Calculator
Find out how many ways you can arrange or choose items from a group. This tool gives you the exact number, plus a full breakdown so you understand how it was calculated.
Permutation
Order matters here. Swapping two items creates a brand new result.
Example. Ranking three runners in first, second and third place.
Combination
Order does not matter here. Swapping two items gives the same result.
Example. Picking three toppings for a pizza.
What is a permutation?
A permutation counts how many ways you can arrange items when the order matters. Picture three friends lining up for a photo. Putting Sam first is a different outcome than putting Sam last, even though the same three people are in the photo.
n is the total number of items you are choosing from.r is how many of those items you are arranging.! means factorial. It means multiplying a number by every whole number below it, down to 1.
What is a combination?
A combination counts how many ways you can choose items when the order does not matter. Picking a team of 3 people from a group of 10 gives the same team no matter what order you picked them in.
n is the total number of items available.r is how many items you are selecting.This formula is the permutation formula divided by
r!, since it removes all the duplicate orderings that permutations count separately.
What does factorial actually mean?
Factorial sounds technical, but it is simple once you see it worked out. The factorial of a number is that number multiplied by every whole number smaller than it, all the way down to 1.
Why does repetition change the formula?
Sometimes an item can be picked more than once. Choosing digits for a 4 digit PIN code is a good example, since any digit can repeat. This changes the math completely, since repeated outcomes now count as valid results.
r positions can be filled with any of the n items, including ones already used. A 4 digit PIN using digits 0 to 9 gives 10,000 possible codes.
Which one should you use?
Ask yourself one question first. Does changing the order create a different result? If yes, use permutation. If the group itself is all that matters, use combination.
Frequently asked questions
Ask if rearranging the same items creates a new outcome. A race podium is a permutation, since first and second place are different. A committee of members is a combination, since the group is the same no matter the order they joined.
Not without repetition allowed. You cannot arrange 5 items into 8 positions if each item can only be used once. Turn on repetition if items can repeat.
Permutations count every possible order as a separate result. Combinations group all those matching orders into one single result, which is why the number is always smaller or equal.
Password strength, lottery odds, scheduling, seating charts, and team selection all rely on this math. It also shows up in probability, which builds directly on these same counting rules.
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