Permutation & Combination Calculator
Enter n and r and you'll get both nPr (permutations, where order matters) and nCr (combinations, where it doesn't) at the same time. Handy for probability problems, counting arrangements, or checking your manual calculation.
How to use
- Enter n (the total number of items) and r (how many you're choosing).
- You get nPr automatically - that's the number of arrangements when order matters, like ranking.
- nCr is shown too - that's combinations, where you just care about which items are picked, not the order.
Related tools:
Permutations count ordered arrangements; combinations count unordered selections. If you are arranging 3 books out of 5 on a shelf, order matters (A-B-C is different from C-B-A), so you use permutations: 5P3 = 5!/(5-3)! = 60. If you are choosing a 3-person committee from 5 candidates, order does not matter (A-B-C and C-B-A are the same team), so you use combinations: 5C3 = 5!/(3! x 2!) = 10.
The key question is always: does order matter? Lottery numbers (pick 6 from 49): order does not matter - combinations. Pin code arrangements: order matters - permutations. Card hand from a deck: usually combinations unless specific order of draw matters. Once you identify which applies, the formulas follow directly.
Frequently Asked Questions
What is the factorial and why does it appear in both formulas?
n! (n factorial) means n x (n-1) x (n-2) x ... x 1. It counts the total ways to arrange n distinct objects. Permutation and combination formulas divide by the arrangements we want to exclude - for combinations, we divide by r! to eliminate the ordering of the selected items.
What happens when n equals r in permutations?
nPn = n! - you are arranging all n items, giving the total number of distinct orderings. For 4 items: 4P4 = 24 arrangements.
Can nCr ever equal nPr?
Only when r = 0 or r = 1. When r = 0, both equal 1. When r = 1, both equal n. For r >= 2, nPr is always larger than nCr because nPr = nCr x r!.