Permutations & Combinations (nPr & nCr)
Calculate total possible ordered permutations ($nPr$) and unordered combinations ($nCr$) with step-by-step factorial breakdowns.
336
Formula: n! / (n - r)!
56
Formula: n! / [r! * (n - r)!]
Detailed Factorial Step Breakdown:
What are Permutations and Combinations?
Combinatorics is the branch of mathematics dealing with counting possibilities. A Permutation ($nPr$) is an arrangement of items where the sequence or order is strictly important (such as a 4-digit PIN code or the podium finishes in a marathon). A Combination ($nCr$) counts groupings where order is irrelevant (such as choosing 3 toppings for a pizza or selecting 5 lottery balls).
Key Capabilities
- Dual nPr & nCr Calculation: Simultaneously calculates ordered permutations and unordered combinations.
- Arbitrary Precision Factorials: Uses JavaScript BigInt mathematics to prevent integer overflow errors on large sets.
- Step-by-Step Factorial Simplification: Shows cancellation steps and factorial expansions clearly.
- Input Boundary Validation: Automatically validates that $n geq r geq 0$ to prevent undefined domain values.
How to Use the nPr / nCr Calculator
- Enter the total population count ($n$).
- Enter the sample size of items being chosen ($r$).
- View the calculated nPr and nCr results instantly alongside the complete step-by-step formula.
Frequently Asked Questions
Why is nPr always greater than or equal to nCr?
Because permutations count every single reordering of the same group as a separate distinct outcome. Specifically, $nPr = nCr imes r!$. Since $r! geq 1$, permutations will always equal or exceed combinations.
What is 0! (zero factorial)?
By mathematical convention and definition, $0! = 1$. This ensures formulas like $nPn = n! / (n - n)! = n! / 1 = n!$ remain universally consistent.
Can r be greater than n?
Without repetition, you cannot choose more items than exist in the set ($r$ cannot exceed $n$). If $r > n$, the number of possible outcomes is 0.
Is this tool free?
Yes, it is 100% free with unlimited calculations.