Permutation and Combination Calculator

Given the sample size, a permutation is the number of ways a certain number of objects can be arranged in sequential order, while a combination is the number of ways a certain number of items can be grouped together. Both feed straight into probabilities. Ever wondered about your chances of winning a lottery? You need the correct numbers in the right sequence, which is pure permutation thinking; a similar concept stretches to combinations too.

This calculator handles one of the most typical concepts of permutations: arrangements of a fixed number of elements r taken from a given set n. You'll see it written as r-permutations of n or partial permutations, denoted nPr, P(n,r), among others. Working without replacement, every way the elements can be listed in a particular order counts, but the number of choices reduces each time an element is chosen, unlike a combination lock where a value can occur multiple times like 3-3-3. Picture picking a team captain and goalkeeper from a soccer team of 11 members, labeled A through K. They cannot be the same person, so once A is chosen, A is removed from the set. If every position mattered, the total possibilities would be 11 × 10 × 9 × 8 × 7 × ... × 2 × 1, or 11 factorial, written 11!. But only the first two choices matter here, 11 × 10 = 110, so the equation removes the rest, 9 × 8 × 7 × ... × 2 × 1, or 9!. That gives the generalized equation below. For the curious, permutations with replacement follow nPr = nr instead.

Strip away the lock and think of a race: out of three students A, B, and C, how many ways can they come 1st, 2nd, 3rd? Because the order is crucial, ABC is different from ACB. Here n is the total number of elements, r is the number of elements to arrange, and "!" is the factorial, the product of all positive integers up to that number. The calculated value is the unique ways you can arrange r objects from a set of n objects.

Combinations are really permutations with all the redundancies removed, since order is not important. They're denoted nCr, C(n,r), or most commonly the binomial coefficient written as (n over r), and again the case here is without replacement, not with replacement. Back to the soccer team: find the number of ways to choose 2 strikers from a team of 11. Unlike picking the captain first then the goalkeeper, the order does not matter because both end up strikers, so whether it's A and then B, or B and then A, only that they're chosen counts. The arrangements for all n people is n!, exactly as in the permutations section. To get the combinations, you remove the redundancies from the total number of permutations (the 110 from earlier) by dividing out 2!, since order no longer matters and A then B equals B then A, that's 2 ways, or 2!. So the permutation equation gets reduced by its redundancies, yielding the formula below. Naturally there are fewer choices for a combination than a permutation. For the curious, combinations with replacement run nCr = (r + n − 1)! / (r! × (n − 1)!).

Selection of items where the order doesn't matter shows up everywhere, like forming a team of three out of students A, B, and C, where ABC is the same as ACB because order doesn't influence who is part of the team. That's the different formula at work, nCr = n! / [(r!) * (n − r)!], where n and r mean the same as before, returning the total number of ways to choose r objects from a set of n objects without considering order.

A couple of worked examples make the calculation concrete, and the steps stay the same no matter how large the numbers get. If the formula still feels confusing, just use our calculator, it even surfaces the permutation and combinations examples for you.

With the definitions settled, the contrast between permutations and combinations comes down to two main differences. First, because permutation calculates the number of possible ways to arrange a certain number of items, different sequences of the same items are considered different, so ABC and BCA are two different permutations, whereas for combination they're considered the same. Second, the two solve different probability problems: permutation deals with sequential problems like the lottery, while combinations solve problems that ignore sequence.

For anyone pushing past the basics, you can add Rules that reduce the List. The "has" rule says certain items must be included for an entry to be included, so has 2,a,b,c means an entry must hold at least two of the letters a, b and c. The "no" rule means some items must not occur together, so no 2,a,b,c means an entry must not have two or more of a, b and c. The "pattern" rule lets you impose a pattern on each entry, so pattern c,* means the letter c must be first and anything else can follow. Drop each rule on its own line. As a quick demo, a,b,c,d,e,f,g has 2,a,b returns the combinations that carry at least 2 of a, b or c.