Permutations vs Combinations: When Order Matters
There is one question that decides which you need: does the order matter? The Permutation and Combination Calculator does both.
The distinction, in one example
Three people from a group of ten.
If you are picking a first, second and third prize, order matters - A then B then C is a different outcome from C then B then A. That is a permutation, and there are 720 of them.
If you are picking three people for a committee, order does not matter - the same three people are the same committee however you listed them. That is a combination, and there are 120.
The ratio between those two answers is exactly 6, which is 3 factorial - the number of ways to order three things. That relationship always holds: permutations equal combinations times r factorial, because every group of r can be arranged r! ways.
The formulas
Permutations, written nPr, are n! divided by (n-r)!. For 10P3: 10! over 7!, which is 10 times 9 times 8, equals 720.
Combinations, written nCr, are n! divided by r! times (n-r)!. For 10C3: that same 720 divided by 3!, equals 120.
Notice the factorials cancel enormously - you never actually compute 10!, you compute 10 times 9 times 8. Which is why these numbers stay manageable even though the factorials inside them do not.
Combinations are symmetrical: nCr equals nC(n-r). Choosing 3 from 10 is the same count as choosing which 7 to leave out, which is often the faster calculation.
Telling them apart in practice
Ask what would change if you swapped two selections round.
Order matters - a podium, a password, a PIN, a queue, a ranking, the batting order, which door you try first.
Order does not matter - a committee, a hand of cards, a pizza's toppings, a lottery ticket, which three books you take on holiday.
The giveaway words are usually "arrange", "order" and "rank" for permutations, and "choose", "select" and "group" for combinations.
A national lottery picking 6 from 49 is a combination - the balls come out in some order and nobody cares. That is 13,983,816 tickets, which is the odds of matching all six.
The repetition question
Both formulas above assume no repetition - each item is used at most once. A four-digit PIN allows repeats, so it is not a permutation in this sense: it is simply 10 to the power 4, which is 10,000.
Deciding whether repetition is allowed is as important as deciding whether order matters, and the two questions together cover every basic counting problem.
Common questions
Which is always the larger number?
Permutations, except when r is 1 or 0, where they are equal.
Is nCr the same as the binomial coefficient?
Yes, the "n choose r" in Pascal's triangle and the binomial theorem.
How do I count arrangements with repeated items?
Divide the total arrangements by the factorial of each repeated group's size - MISSISSIPPI has fewer distinct arrangements than 11 letters would suggest.
Got a counting problem?