Permutation & Combination Calculator (nPr / nCr)

Compute both nPr (permutations — arrangements where order matters) and nCr (combinations — selections where it doesn't) in one go. Results are exact even for large values, using big-integer arithmetic.

How to use this calculator

Enter n — how many items you have in total.
Enter r — how many you're arranging or choosing.
Press Calculate to get both nPr and nCr, computed exactly.
Ask yourself: does order matter? Podium finishes → permutation. Lottery numbers → combination.

Formula used

nPr = n! ÷ (n − r)! · nCr = n! ÷ (r! × (n − r)!)

n! (factorial) means n × (n−1) × … × 1. Permutations count ordered arrangements; dividing by r! removes the orderings of the chosen items, giving combinations. That's why nPr = nCr × r! always.

Example calculation

Worked example

From 10 runners, how many ways can 3 medals (gold, silver, bronze) be awarded? Order matters → P(10,3) = 10 × 9 × 8 = 720.

How many ways to pick any 3 of the 10 for a team? Order irrelevant → C(10,3) = 720 ÷ 3! = 720 ÷ 6 = 120.

Permutations vs combinations

Both count ways of choosing r items from n — the difference is whether order matters. A permutation treats ABC and CBA as different (passwords, rankings, seatings); a combination treats them as the same selection (committees, lottery tickets, pizza toppings).

Since every unordered selection of r items can be internally ordered r! ways, permutations always exceed combinations by exactly that factor. These counts grow explosively — C(49,6) is nearly 14 million, which is precisely why lottery jackpots are so hard to win.

Why use this calculator?

Get exact answers even for large n, where ordinary calculators overflow.
See both counts side by side with the r! ratio that connects them.
Settle the order-matters question with the medal-vs-team framing.

Frequently asked questions

How do I know whether to use nPr or nCr?

Ask if swapping two chosen items creates a different outcome. Different → order matters → permutation (nPr). Same → combination (nCr). Assigning roles is nPr; forming a group is nCr.

What is 0! and why does nC0 equal 1?

By definition 0! = 1 — it makes the formulas consistent. nC0 = 1 because there is exactly one way to choose nothing, and nCn = 1 because there's one way to take everything.

What are the chances of winning a 6-from-49 lottery?

C(49,6) = 13,983,816 possible tickets, so one ticket has a 1 in ~14 million chance ≈ 0.0000072%. Enter n=49, r=6 above to verify.