Permutations (nPr) & Combinations (nCr) Calculator

Count ordered arrangements and unordered subsets from a set of n elements.

Unordered Combinations C(n, r)
495 combinations
Ordered Permutations P(n, r) 11,880 combinations
With Replacement C(n+r−1, r) 1,365 combinations
Subsets of Full Set (2^n) 4,096 combinations
Inputs
Total distinct items in the set (n).
Number of items selected (r).
Results & Graph
Unordered Combinations C(n, r)
495 combinations
Ordered Permutations P(n, r)
11,880 combinations
With Replacement C(n+r−1, r)
1,365 combinations
Subsets of Full Set (2^n)
4,096 combinations

Unordered Combinations C(n, r)

Graph 1 • Total N Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 combinations 1.7k combinations 3.3k combinations 50%: 15 combinations 15 50% 75%: 126 combinations 126 75% 100%: 495 combinations 495 100% 125%: 1,365 combinations 1.4k 125% 150%: 3,060 combinations 3.1k 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Unordered Combinations C(n, r) 495 combinations Ordered Permutations P(n, r) 11,880 combinations With Replacement C(n+r−1, r) 1,365 combinations Subsets of Full Set (2^n) 4,096 combinations
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Unordered Combinations C(n, r)495 combinationsOrdered Permutations P(n, r)11,880 combinationsWith Replacement C(n+r−1, r)1,365 combinationsSubsets of Full Set (2^n)4,096 combinations
Summary: Choosing 4 from 12 yields 495 combinations (nCr) and 11,880 ordered permutations (nPr).
Summary: Choosing 4 from 12 yields 495 combinations (nCr) and 11,880 ordered permutations (nPr).

Formula & Mathematical Methodology: nCr & nPr Mathematical Identity

nCr = n! / [ r!(n − r)! ]
nPr = n! / (n − r)!
nCr = n! / [ r!(n − r)! ]; nPr = n! / (n − r)!
  • totalN: Total N (numeric parameter)
  • chooseR: Choose R (numeric parameter)

Rounding: All calculations run in double-precision IEEE-754 arithmetic; displayed outputs round to up to 4 decimal places.

Worked Example: Worked Mathematical Verification: nCr & nPr

Count ordered arrangements and unordered subsets from a set of n elements.

  1. Verify the domain constraints of the input parameters.
  2. Apply the governing mathematical identity: nCr = n! / [ r!(n − r)! ]; nPr = n! / (n − r)!.
  3. Cross-check the primary value against the supporting invariants and coordinate graph.
Result: Evaluates nCr & nPr via nCr = n! / [ r!(n − r)! ]; nPr = n! / (n − r)!.

Frequently Asked Questions

What mathematical formula does the Permutations (nPr) & Combinations (nCr) Calculator use?

It directly evaluates nCr = n! / [ r!(n − r)! ]; nPr = n! / (n − r)! in double-precision floating-point arithmetic and updates all supporting invariants and graphs live.

How can I compare different parameter sets?

Click Compare in the results toolbar to pin up to three parameter combinations side by side.

Can I export the mathematical report or share a link?

Yes. Use the Share Link, Copy Summary, or Markdown/JSON export buttons in the results panel.

Mathematics & Statistics • Combinatorics Guide

How to Use the Permutations (nPr) & Combinations (nCr) Calculator: Practical Guide & Visual Analysis

Calculate combinations C(n,r), ordered permutations P(n,r), and multiset combinations with replacement. Includes textbook mathematical notation, step-by-step verification, and three custom SVG visualizations. Whether you are researching permutations (npr) & combinations (ncr) calculator, ncr & npr, permutations combinations factorial, combinatorics calculator, this interactive tool helps you calculate combinations C(n,r), ordered permutations P(n,r), and multiset combinations with replacement.

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust Total N, Choose R, the calculator immediately computes Primary exact or numerical solution, Supporting identities & invariants, 3-Panel mathematical SVG plots and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Universal Mathematical & Statistical Standard (ISO 80000-2))

In our benchmark scenario (Worked Mathematical Verification: nCr & nPr), Evaluates nCr & nPr via nCr = n! / [ r!(n − r)! ]; nPr = n! / (n − r)!. Keep in mind that uses standard Euclidean geometry, real/complex algebra, and NIST statistical definitions. For related calculations in Mathematics & Statistics, explore Scientific Calculator, Statistics & Standard Deviation Calculator, Geometry Calculator (Right Triangle, Circle & Cylinder).