Permutation & Combination Calculator
Permutations (nPr)720
Combinations (nCr)120
Permutations and combinations count how many ways you can select items from a set. A permutation (nPr) counts arrangements where order matters, while a combination (nCr) counts selections where order is irrelevant. This calculator computes both at once from the total number of items n and the number chosen r.
Formula
nPr = n! / (n - r)!; nCr = n! / (r! (n - r)!)
- n
- Total number of distinct items available
- r
- Number of items selected (0 ≤ r ≤ n)
- !
- Factorial, the product of all positive integers up to that number
How it works
- Enter the total number of items n and the number you are choosing r. Both must be whole numbers with r no larger than n.
- The calculator returns nPr, the number of ordered arrangements, and nCr, the number of unordered selections. Because every arrangement is one of r! orderings of a combination, nPr is always r! times larger than nCr.
Worked example
Choosing 3 items from a set of 10, counting both ordered and unordered selections.
- nPr = 10 × 9 × 8 = 720 ordered arrangements.
- nCr = 720 / 3! = 720 / 6 = 120 unordered selections.
nPr = 720 permutations and nCr = 120 combinations.
Frequently asked questions
- When should I use a permutation versus a combination?
- Use a permutation (nPr) when the order of selection matters, such as assigning first, second, and third place. Use a combination (nCr) when order does not matter, such as choosing a committee of three people.
- Why must r be less than or equal to n?
- You cannot choose more items than exist in the set, so r cannot exceed n. If r is larger than n, or either value is negative or not a whole number, the calculator returns no result.
- How are nPr and nCr related?
- They differ by the number of orderings of the chosen items: nPr = nCr × r!. Each unordered combination of r items can be arranged in r! different orders, so there are always more permutations than combinations.