Permutation and Combination Calculator
nPr & nCr Formula Breakdown
Understanding Permutations & Combinations
Both count ways to select r items from a set of n — the only question is whether the order of selection matters.
Think of a race with 8 runners and 3 medal positions — who gets gold, silver, and bronze is a different outcome for every possible order. That's a permutation problem: nPr.
Think of choosing 3 people from 8 to form a committee — there are no roles, so the group {Alice, Bob, Carol} is the same regardless of pick order. That's a combination problem: nCr.
The Formulas
Permutations
Combinations
Why Combinations Are Always Smaller
Every combination of r items can be rearranged into r! different permutations. That's exactly why the combination formula divides the permutation formula by an extra r! — it collapses all those equivalent orderings into a single count. For example, choosing 3 letters from {A, B, C, D} as a combination gives 4 results, but as a permutation (where ABC and CBA count separately) it gives 24 — six times as many, since 3! = 6.
Permutations are always at least as large as combinations.
Zero factorial is defined as 1 by convention.
You cannot choose more items than exist in the set.
Key Takeaways
- Order matters for permutations, not for combinations. That's the entire distinction.
- Both n and r must be non-negative whole numbers, and r can never exceed n.
- nCr is the building block of combinatorial probability — see the Probability Calculator's Combinatorial tab.
- Large n values grow factorials extremely fast — double-check that you actually need the full permutation count rather than the smaller combination count.
Frequently Asked Questions
- Enter n, the total number of items.
- Enter r, the number of items being chosen or arranged.
- Click "Calculate" to see both nPr (permutations) and nCr (combinations), with the full factorial breakdown for each.
A permutation (nPr) counts arrangements where order matters — "first place, second place, third place" is a different outcome than "third, second, first." A combination (nCr) counts selections where order doesn't matter — picking 3 people for a committee is the same group regardless of the order you picked them in. Since every combination can be arranged in r! different orders, permutations are always greater than or equal to combinations for the same n and r.
nPr = n! ÷ (n − r)! starts with all n! orderings of the full set, then divides out the orderings of the n − r items you didn't choose (since their internal order doesn't affect which r items were picked and in what order). nCr = n! ÷ (r!(n − r)!) takes that same permutation count and additionally divides by r! to collapse every reordering of the chosen r items into a single combination.
You can't choose or arrange more items than exist in the set — if you only have 5 items, you can't select 7 of them. The calculator rejects any input where r > n since it's mathematically undefined for both permutations and combinations in the ordinary sense.
Both nP0 and nC0 equal 1 — there's exactly one way to choose or arrange zero items (do nothing). nCn also equals 1 — there's only one way to choose every item. nPn = n!, since arranging all n items in every possible order is simply every permutation of the full set.
Yes. Both permutations and combinations are defined using factorials, which only make sense for non-negative integers. The calculator rejects decimals and negative numbers for either field.