Permutations & Combinations Calculator
Calculate nPr (order matters) and nCr (order does not matter).
Does the order matter?
One question separates the two. Permutations count arrangements where order matters; combinations count selections where it does not. Choosing a president and a treasurer from ten people is a permutation, because who holds which role changes the outcome. Choosing two people for a committee is a combination.
The formulas differ by one factor: P(n,r) = n!/(n−r)! andC(n,r) = n!/(r!(n−r)!). That extra r! divides out the orderings you have decided not to distinguish, which is why there are always fewer combinations than permutations — P(10,2) is 90 while C(10,2) is 45, exactly half, because each pair can be ordered two ways.
Both assume selection without replacement. With replacement the count is simplynʳ, which is how a four-digit PIN gives 10,000 possibilities rather than a permutation count. And 0! = 1 by convention, not by accident — there is exactly one way to arrange nothing, and the definition keeps every formula consistent at the boundary.
Frequently Asked Questions
What is the difference between Permutation and Combination?
In permutations (nPr), order matters. In combinations (nCr), order does not matter.
What are the formulas for nPr and nCr?
nPr = n! / (n - r)! and nCr = n! / (r! × (n - r)!).
What is the difference between a permutation and a combination?
Order matters in permutations and does not in combinations. Choosing 3 people from 5 for president, secretary and treasurer is a permutation (60 ways); choosing 3 for an unranked committee is a combination (10 ways). Combinations are always fewer, by a factor of r!.
What are the formulas for nPr and nCr?
P(n,r) = n! / (n-r)! and C(n,r) = n! / (r! x (n-r)!). The combination formula is the permutation divided by r!, which removes the double-counting of the same group in different orders. Both require r to be no greater than n.