Permutation & Combination Calculator
Calculate nPr permutations and nCr combinations
About the Permutation & Combination Calculator
When you need to count arrangements or selections, the difference between a permutation and a combination is everything — and it is easy to mix them up. This calculator computes both at once from the same two inputs: n, the total number of items, and r, how many you are choosing.
Permutations (nPr) count ordered arrangements — useful for problems like how many ways a race of n runners can finish in the top r places. Combinations (nCr) count unordered selections — how many r-person committees you can form from n people, or how many lottery tickets cover a draw. Students checking probability homework and anyone setting up a counting problem get both answers plus the factorial breakdown behind them.
How to Use the Permutation & Combination Calculator
- Enter n, the total number of items, in the first field.
- Enter r, the number of items being chosen, in the second field.
- Read the permutation (nPr) and combination (nCr) results, both calculated automatically.
- Expand the factorial breakdown to see n!, r!, and (n−r)! and how they combine.
- Use Copy to save both formulas and values, or Reset to clear and start over.
Why Use ToolForge’s Permutation & Combination Calculator
- It computes both nPr and nCr simultaneously, so you can compare ordered and unordered counts side by side instead of running two tools.
- Each answer is shown with its formula and a factorial breakdown — n!, r!, and (n−r)! — making it a genuine learning aid, not just a black-box number.
- It validates the inputs, returning zero when r exceeds n or when a value is negative, so impossible cases do not produce misleading output.
- Large results are formatted with thousands separators and computed instantly in the browser, with no sign-up.
Frequently Asked Questions
What is the difference between a permutation and a combination?
A permutation counts arrangements where order matters (nPr); a combination counts selections where order does not (nCr). Choosing a president and vice-president from a group is a permutation; choosing two committee members is a combination. For the same n and r, there are always at least as many permutations as combinations.
What does the factorial notation n! mean?
n! ("n factorial") is the product of all positive integers up to n — for example 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition 0! and 1! both equal 1. Factorials are the building blocks of the permutation and combination formulas shown in the breakdown.
Why do I get 0 as a result?
The calculator returns 0 when the inputs describe an impossible selection — most commonly when r is larger than n, or when either value is negative. You cannot choose more items than exist, so check that r is no greater than n.
What are the formulas used?
Permutations use nPr = n! ÷ (n−r)!, and combinations use nCr = n! ÷ [r! × (n−r)!]. The tool displays both formulas with your numbers substituted in so you can verify each step.
