Mathematics · Probability
Combinations and Permutations Calculator
Count selections when order either matters or does not matter.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Permutation count = 5! ÷ (5 − 3)! = 60.
- Combination count divides by 3! because order is ignored.
- Combination count = 10.
Definition and formula
Permutations count ordered arrangements; combinations count selections where rearranging the same chosen items creates no new outcome.
nPr = n! ÷ (n − r)!; nCr = n! ÷ [r!(n − r)!]
Worked example
Choosing 3 of 5 gives 10 combinations but 60 ordered permutations.
Common mistake
Use permutations only when changing the order creates a distinct outcome.
How to learn with this calculator
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Clear answers
Frequently asked questions
What does the Combinations and permutations do?
Count selections when order either matters or does not matter.
How does the Combinations and permutations work?
The calculator applies nPr = n! ÷ (n − r)!; nCr = n! ÷ [r!(n − r)!]. Permutations count ordered arrangements; combinations count selections where rearranging the same chosen items creates no new outcome.
What can I learn from the Combinations and permutations?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed 2026-07-14. Calculations tested 2026-07-14.