Permutation & Combination Calculator
nPr counts ordered arrangements, nCr counts unordered selections — the two counting-problem answers that constantly get swapped for each other, resolved together from the same n and r so you can see both at once.
Inputs
Result
nPr = 720
nCr = 120
How the permutation & combination calculator works
Permutations (order matters): nPr = n! ÷ (n−r)!.
Combinations (order doesn't matter): nCr = nPr ÷ r! = n! ÷ (r! × (n−r)!).
Combinations are always smaller than or equal to permutations for the same n and r, since every combination corresponds to r! different orderings that permutations count separately.
Worked example: n = 10, r = 3
- nPr = 10! ÷ 7! = 10 × 9 × 8 = 720.
- nCr = 720 ÷ 3! = 720 ÷ 6 = 120.
- 720 ordered arrangements collapse into 120 unordered groups because each group of 3 can be ordered 3! = 6 different ways.
Common mistakes to avoid
Using permutations when order genuinely doesn't matter
Picking 3 people for a committee (roles undefined) is a combination problem; picking a president, vice-president and treasurer from the same group is a permutation problem — the same n and r give very different answers depending on which applies.
Entering r larger than n
You can't choose or arrange more items than exist in the set — r must be less than or equal to n, or the calculation is meaningless.
Frequently asked questions
How do I know whether my problem needs permutations or combinations?
Ask whether swapping two selected items would create a genuinely different outcome. If yes (like assigning distinct roles), use permutations; if no (like selecting an unordered group), use combinations.
What does nCr = 1 mean when r equals n?
There is only one way to choose all n items from a set of n — the entire set itself, with nothing left to choose between.
Why is nCr always the smaller of the two results?
Every unordered selection of r items corresponds to r! different ordered arrangements, so combinations divide permutations down by that overcounting factor.
Is this related to the binomial probability formula?
Yes directly — nCr is exactly the coefficient used in the Binomial Probability Calculator to count how many ways k successes can occur among n trials.
Related calculators
Binomial Probability Calculator
Probability of exactly k successes in n trials.
Factorial Calculator
Compute n! for any reasonable value of n.
Probability Calculator
Single and combined event probability.
Statistics Calculator
Mean, median, mode and standard deviation of a data set.
Exponent Calculator
Raise any base to any power.