Permutation & Combination Calculator
Calculate nPr and nCr — permutations with repetition, combinations with repetition — with exact, unrounded results for any n and r.
720
10P3
120
10C3
Choose n and r
Permutation 10P3
720
Order matters, no repetition
Combination 10C3
120
Order doesn’t matter, no repetition
720
nPr (no repeat)
120
nCr (no repeat)
1,000
Perm w/ repeat
220
Comb w/ repeat
All Counting Methods — n=10, r=3
Permutation (no repeat)
nPr = n! / (n-r)!
720
Order matters. Arrangements of r from n items.
Combination (no repeat)
nCr = n! / (r!(n-r)!)
120
Order doesn't matter. Selections of r from n.
Permutation with repetition
n^r
1,000
Order matters, repetition allowed — e.g. PIN codes.
Combination with repetition
C(n+r-1, r)
220
Order doesn't matter, repetition allowed — e.g. ice cream scoops.
Permutations vs. Combinations: The One Question That Matters
The single most important question when counting arrangements is: does the order matter? If rearranging the same items produces a different outcome, you’re counting permutations— think of a race, where finishing 1st, 2nd, 3rd is a completely different result from finishing 2nd, 1st, 3rd, even with the identical three runners. If rearranging the items changes nothing about the outcome, you’re counting combinations — think of choosing a 3-person committee, where the group {Alice, Bob, Carol} is the same committee no matter what order you list the names in. Every mode below builds on that one distinction, with a second variable layered on top: whether an item can be chosen more than once (repetition allowed) or only once (no repetition).
All Four Counting Formulas
Permutation (no repeat): nPr = n! / (n−r)! — order matters, each item used at most onceCombination (no repeat): nCr = n! / (r!(n−r)!) — order doesn’t matter, each item used at most oncePermutation with repetition: n^r — order matters, items can repeat (e.g. PIN codes)Combination with repetition: C(n+r−1, r) — order doesn’t matter, items can repeat (“stars and bars”)A benchmark worth memorizing: a 6-from-49 lottery draw is a combination — the balls come out in some order, but a ticket wins regardless of the sequence they were drawn in. Plugging n=49 and r=6 into the no-repetition combination formula gives exactly 13,983,816 possible tickets, which is why the odds of matching all six numbers are famously about 1 in 14 million.
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Permutation Example
10 runners, top 3 podium spots: 10P3 = 10 × 9 × 8 = 720 possible podium orders — swapping gold and silver creates a different result.
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Combination Example
10 candidates, a 3-person panel: 10C3 = 720 / 3! = 120 possible panels — the same three people form one panel no matter what order they’re chosen in.
— FAQ