Combinatorics
Combinations vs Permutations: nCr and nPr
Updated October 2, 2026
Published by SolveCalc (Operator of solvecalc.live)
Permutations count arrangements where order matters. Combinations count selections where order does not. Mixing them up is one of the fastest ways to miss a probability or counting question.
Quick decision rule
- Does the order of the items change the outcome? → use a permutation (nPr).
- Is it only about which items are in the group? → use a combination (nCr).
Formulas
With n distinct items, choosing r of them:
- nPr = n! ÷ (n − r)!
- nCr = n! ÷ (r!(n − r)!)
Notice nCr = nPr ÷ r!. Dividing by r! removes the different orders of the same group.
Worked examples
Permutation
How many ways can three students stand in a line of five available people? Order matters. Compute 5P3 = 5 × 4 × 3 = 60.
Combination
How many ways can you choose a committee of 3 from 5 people? Order does not matter. Compute 5C3 = 10.
| Question type | n | r | Result |
|---|---|---|---|
| Line up 3 of 5 (order matters) | 5 | 3 | 5P3 = 60 |
| Choose 3 of 5 (order ignored) | 5 | 3 | 5C3 = 10 |
On CalcSolver, use the nPr and nCr keys on the scientific calculator.
Factorials grow fast
5! = 120, but large factorials overflow ordinary calculator range (CalcSolver shows Infinity for values past the floating-point limit, such as 171!). Keep r and n modest when you experiment.
Related reading
Key layouts for nCr, nPr, and ! are covered in How to use a scientific calculator. For counting with primes and factors, try the Prime Number Checker and LCM and GCF Calculator.