Combination & Permutation Calculator
Enter n (total items) and r (items chosen) to calculate both the number of combinations (order doesn't matter) and permutations (order matters) at once.
How the Combination & Permutation formula works
The combination and permutation formulas:
nPr = n! / (n − r)! nCr = n! / (r! × (n − r)!)
Permutations count every distinct ordering, while combinations group orderings of the same items together, which is why nCr is always less than or equal to nPr.
Step-by-step calculation
- Calculate n factorial (n!) — the product of all whole numbers from 1 to n.
- Calculate (n − r)! the same way.
- For permutations, divide n! by (n − r)!.
- For combinations, additionally divide that result by r! to remove duplicate orderings of the same chosen items.
Worked example
Choosing 3 items from 5 (n=5, r=3): nPr = 5!/(5−3)! = 120/2 = 60 distinct orderings. nCr = 60 / 3! = 60/6 = 10 distinct groups, regardless of order.
Frequently asked questions
When should I use a combination instead of a permutation?
Use combinations when order doesn't matter, like picking a group of 3 people from 10 for a committee. Use permutations when order matters, like awarding 1st, 2nd, and 3rd place among the same 10 people.
Why does the calculator cap how large n can be?
Factorials grow extremely fast — 170! already exceeds what standard double-precision numbers can represent accurately, so extremely large values of n are capped to keep results numerically reliable.