What's the difference between nCr and nPr?

nCr counts selections where order is irrelevant (a hand of 3 cards is the same hand no matter what order you were dealt them). nPr counts arrangements where order matters (1st, 2nd, 3rd place are different outcomes even with the same 3 people). nPr is always ≥ nCr for the same n and r.

Why not just compute n! directly?

Factorials grow enormously fast — 70! already exceeds what a standard number can represent precisely. This calculator multiplies and divides incrementally instead of computing full factorials, so it stays accurate for much larger values of n.

What happens if r is larger than n?

There's no valid way to choose or arrange more items than exist in the set, so both nCr and nPr are 0 whenever r exceeds n.

There's exactly one way to choose or arrange zero items — the empty selection — so both nCr and nPr equal 1 when r is 0, regardless of n.

Where does this come up outside of math class?

Combinations and permutations show up in probability problems, lottery odds, password/PIN possibilities, scheduling and seating arrangements, and card game odds calculations.

Math & Statistics Cluster

Combination & Permutation Calculator

Calculate nCr combinations and nPr permutations.

TL;DR: Combination & Permutation Calculator is a free, browser-based tool that lets you calculate nCr combinations and nPr permutations.

n (total items)

r (items chosen)

About the Combination & Permutation Calculator

The Combination & Permutation Calculator solves mathematical equations, statistical distributions, and numerical relationships with step-by-step arithmetic breakdowns and high-precision calculations.

How to Use the Combination & Permutation Calculator

Enter your known values, dataset numbers, or mathematical coefficients.
Select your preferred precision decimal places or calculation mode.
Click to compute and review the step-by-step derivation steps.
Copy the solved numerical answer or explore related mathematical proofs.

Mathematical Formula & Methodology

Here is the standard mathematical equation used to calculate this result accurately:

nCr = \frac{n!}{r!(n-r)!} \quad (\text{Combinations}), \quad nPr = \frac{n!}{(n-r)!} \quad (\text{Permutations})

Formula Variables Explained:
Calculates the number of possible unordered subsets (combinations) and ordered arrangements (permutations) of $r$ items chosen from $n$ items.

Step-by-Step Worked Example

Choosing 3 committee members from 10 people: $10C3 = \frac{10!}{3!7!} = \frac{10 \times 9 \times 8}{6} = 120 \text{ combinations}$

Expert Tips for Combination & Permutation Calculator

Sample vs Population Data: In statistical variance and standard deviation, use sample formulas (Bessel's correction $n-1$) when analyzing subset samples.
Order of Operations: Computations strictly adhere to standard mathematical PEMDAS / BODMAS operational precedence.
High Precision Arithmetic: Internal calculations use 64-bit IEEE floating-point arithmetic for extreme numerical accuracy.

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100% Client-Side Privacy Guarantee

All processing runs locally inside your browser using JavaScript and HTML5 APIs. Your data, files, and inputs are never uploaded to any remote server. Complete privacy by design.

Frequently Asked Questions

: nCr counts selections where order is irrelevant (a hand of 3 cards is the same hand no matter what order you were dealt them). nPr counts arrangements where order matters (1st, 2nd, 3rd place are different outcomes even with the same 3 people). nPr is always ≥ nCr for the same n and r.
: Factorials grow enormously fast — 70! already exceeds what a standard number can represent precisely. This calculator multiplies and divides incrementally instead of computing full factorials, so it stays accurate for much larger values of n.
: There's no valid way to choose or arrange more items than exist in the set, so both nCr and nPr are 0 whenever r exceeds n.
: There's exactly one way to choose or arrange zero items — the empty selection — so both nCr and nPr equal 1 when r is 0, regardless of n.
: Combinations and permutations show up in probability problems, lottery odds, password/PIN possibilities, scheduling and seating arrangements, and card game odds calculations.

Cite This Tool

Referencing this tool in a paper, article, or bibliography? Copy a ready-made citation below.

NexLove.org. (2026). Combination & Permutation Calculator [Software]. https://nexlove.org/tools/combination-permutation-calculator.html

“Combination & Permutation Calculator.” NexLove.org, 2026, nexlove.org/tools/combination-permutation-calculator.html.

NexLove.org (2026) Combination & Permutation Calculator. Available at: https://nexlove.org/tools/combination-permutation-calculator.html (Accessed: 2026).

@misc{nexlove_combination_permutation_calculator,
  title = {Combination & Permutation Calculator},
  author = {{NexLove.org}},
  year = {2026},
  url = {https://nexlove.org/tools/combination-permutation-calculator.html}
}

Embed This Tool

Add the live Combination & Permutation Calculator tool to your own website with this snippet — it loads the real, working tool in an iframe, not a static screenshot.

<iframe src="https://nexlove.org/embed/combination-permutation-calculator.html" width="100%" height="600" style="border:1px solid #e2e8f0;border-radius:12px" title="Combination & Permutation Calculator — NexLove.org" loading="lazy"></iframe>

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