Permutation & Combination Calculator
Calculate the number of permutations (nPr) and combinations (nCr) of r items chosen from a set of n.
- nPr — order matters720
- nCr — order doesn't matter120
Every group of 3 items can be arranged 6 different ways, which is exactly why nPr is 6× bigger than nCr here.
- C(10,3) — Combinations120
About the Permutation & Combination Calculator
Permutations and combinations both count the number of ways to choose r items from a set of n, but they answer subtly different questions: permutations count arrangements where order matters (who finishes 1st, 2nd, 3rd in a race), while combinations count selections where order doesn't matter (which 3 people get picked for a committee, regardless of the order they're picked in). Mixing the two up is one of the most common mistakes in an introductory statistics or combinatorics course.
This calculator computes both nPr and nCr for the same n and r side by side, which makes the relationship between them — nPr is always nCr multiplied by the number of ways to arrange r items — much easier to see than working with either formula in isolation.
How it’s calculated
nPr (permutations) = n! / (n−r)!. The factorial n! means n × (n−1) × (n−2) × ... × 1, the product of every whole number from n down to 1. Dividing by (n−r)! effectively 'stops' the multiplication after r terms, which is exactly what's needed since you're only choosing and arranging r items out of the n available.
nCr (combinations) = n! / (r! × (n−r)!) — it's nPr divided again by r!, the number of ways to arrange those same r chosen items among themselves. Dividing by r! removes the effect of order, which is exactly the difference between counting arrangements and counting selections.
Frequently asked questions
What's the difference between a permutation and a combination?
A permutation counts arrangements, where order matters — 1st, 2nd, 3rd place in a race are all different outcomes even with the same three people. A combination counts selections, where order doesn't matter — picking the same 3 people for a committee is one outcome no matter what order they were chosen in.
What does the factorial symbol (!) mean?
n! means multiply every whole number from n down to 1: 5! = 5×4×3×2×1 = 120. It represents the total number of ways to arrange n distinct items in a row, and by convention 0! is defined as 1.
Can you give a real-world example of each?
Permutation: assigning gold, silver, and bronze medals to 3 of 10 racers — the same 3 racers in a different order is a different outcome. Combination: choosing 3 of 10 people for a study group — the same 3 people is the same outcome regardless of who was picked first.
Why does the combination formula divide by an extra r!?
Because nPr already counts every distinct arrangement of the r chosen items separately, and there are exactly r! ways to arrange any given group of r items. Dividing nPr by r! collapses all of those duplicate arrangements back down to a single combination.
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