Calculate permutations (nPr) and combinations (nCr) instantly. Find how many ways to arrange or select items from a set. Free online permutation and combination calculator with formulas, examples, and step-by-step guide.
The calculator computes r-permutations of n, or partial permutations, denoted as nPr. In permutations without replacement, all possible ways that elements can be listed in a particular order are considered, but the number of choices reduces each time an element is chosen.
Choosing a captain and goalkeeper from 11 players (A-K):
Captain: 11 choices โ Goalkeeper: 10 choices (captain removed)
11P2 = 11! / 9! = 11 ร 10 = 110
Only the first two choices matter (11 ร 10); the rest (9!) is removed from the equation.
Combinations are related to permutations in that they are essentially permutations where all the redundancies are removed, since order in a combination is not important. Combinations are denoted as nCr, C(n,r), or most commonly as the binomial coefficient (โฟแตฃ).
Choosing 2 strikers from 11 players. Order does not matter (A then B = B then A).
Remove redundancies by dividing by 2! (the number of ways 2 players can be arranged):
11C2 = 11! / (2! ร 9!) = 55
There are fewer combinations (55) than permutations (110) since redundancies are removed.
Let's start with a super simple example.
Imagine you have three letters: A, B, and C. You want to pick two of them.
That's it. That's the whole difference. Sounds simple, right? But it trips up so many people because real-world problems aren't always obvious.
Using our calculator is stupidly easy. Here's what you do:
You have a lock with digits 0-9 (that's 10 digits total). You need a 3-digit code. Does order matter? Yes! 1-2-3 is different from 3-2-1. So you'd use the permutation calculator. n = 10, r = 3. The answer? 720 possible codes.
You're at a pizza place with 8 toppings. You can pick 3. Does order matter? No! Pepperoni, mushrooms, and olives is the same pizza no matter what order you say them. So you'd use the combination calculator. n = 8, r = 3. The answer? 56 different pizza combinations.
Okay, let's look at the actual math. Don't freak out - it's simpler than it looks.
nPr = n! / (n - r)!
That exclamation mark is a factorial. It just means multiply the number by every positive integer smaller than it. So 5! = 5 ร 4 ร 3 ร 2 ร 1 = 120.
Here's what the formula is really saying: "Start with n items. For the first spot, you have n choices. For the second spot, you have n-1 choices. Keep going until you've picked r items."
nCr = n! / [r! ร (n - r)!]
Notice it's almost the same as the permutation formula. The only difference is we divide by r! at the end. Why? Because in combinations, all the different orders of the same items count as one. Dividing by r! removes those extra arrangements.
Here's a cool trick: nCr = nPr / r! So if you already calculated the permutation, just divide by r! to get the combination.
This is the part everyone struggles with. Here's a simple test:
Ask yourself: "If I swap two items, do I get a different result?"
Yes โ nPr (Permutations)
No โ nCr (Combinations)
I've seen these mistakes over and over on Reddit and in math classes. Here's what to watch out for:
Mistake 1: Using nPr when you should use nCr (or vice versa). Always ask yourself if order matters. If you're not sure, try a tiny example with just 2 or 3 items and see if swapping them changes the result.
Mistake 2: Forgetting that 0! = 1. When r = n, you get (n - r)! = 0! = 1. If you forget this, your calculation will be wrong.
Mistake 3: Putting r bigger than n. You can't pick 5 items from a group of 3. The calculator will give you an error or zero. Always make sure r โค n.
Mistake 4: Confusing permutations with the counting principle. The counting principle is for multiple independent choices (like choosing a shirt AND pants). Permutations are for arranging items from one set.
nCr = nC(n-r). So 10C3 = 10C7. Why? Because choosing 3 items to keep is the same as choosing 7 items to throw away. This can save you time with big numbers.
Our calculator handles permutations without repetition. But what if you can reuse items? Like a lock where digits can repeat? That's just n^r. So for a 3-digit lock with 10 digits, it's 10^3 = 1000.
For things like "how many ways to choose 3 scoops of ice cream from 5 flavors if you can pick the same flavor multiple times?" The formula is C(n+r-1, r).
You might be thinking, "When am I ever going to use this?" More often than you'd think.
Even if you never use it for work, it's a great brain workout. It teaches you to think systematically about counting and probability.
People have been counting combinations for thousands of years. Ancient Indian mathematicians worked with permutations as early as 300 BCE. The modern formulas were developed by French mathematicians Blaise Pascal and Pierre de Fermat in the 1600s while they were figuring out gambling problems. So next time you use our calculator, you're standing on the shoulders of some pretty smart people.
Remember, everyone struggles with this at first. It's not just you. The key is practice and understanding the "order matters" concept.
Permutations care about order. Combinations don't. For example, picking a 3-digit lock code is a permutation because 1-2-3 is different from 3-2-1. Picking 3 pizza toppings is a combination because pepperoni, mushrooms, and olives is the same pizza no matter what order you list them.
Ask yourself: "If I swap two items, do I get a different result?" If yes, use nPr (permutations). If no, use nCr (combinations). For example, swapping 1st and 2nd place in a race gives a different result, so it's a permutation. Swapping two committee members doesn't change the committee, so it's a combination.
nPr stands for "number of permutations of n items taken r at a time." It tells you how many different ways you can arrange r items from a set of n items when order matters. Most scientific calculators have an nPr button.
nCr stands for "number of combinations of n items taken r at a time." It tells you how many different groups of r items you can select from a set of n items when order doesn't matter. It's also called the binomial coefficient.
For small numbers, you can list them out. For larger numbers, use the formula nPr = n! / (n-r)!. For example, 5P3 = 5! / 2! = (5ร4ร3ร2ร1) / (2ร1) = 120/2 = 60. You can also multiply n ร (n-1) ร (n-2) ... until you have r factors.
Use the formula nCr = n! / [r! ร (n-r)!]. For example, 5C3 = 5! / (3! ร 2!) = 120 / (6 ร 2) = 120/12 = 10. A shortcut is to calculate the permutation first, then divide by r!.
You can't pick more items than you have. If you have 5 items, you can't choose 7 of them. The formula requires r โค n. If r > n, the factorial of a negative number isn't defined, so calculators show an error.
The counting principle is for independent choices from different categories (like choosing a shirt AND pants). Permutations are for arranging items from one set. For example, choosing a shirt from 5 options AND pants from 3 options uses the counting principle (5ร3=15). Arranging 3 books on a shelf uses permutations (3P3=6).
Yes, all the time! Probability is often calculated as (number of favorable outcomes) / (total number of possible outcomes). Permutations and combinations help you count those outcomes. For example, the probability of winning the lottery is 1 divided by the number of possible combinations.