What are the mathematical formulas for nCr and nPr?

Combinations: `nCr = n! / [r!(n - r)!]`. Permutations: `nPr = n! / (n - r)!`, where `!` represents the factorial operation.

What is a factorial (n!)?

A factorial is the product of all positive integers less than or equal to n: `n! = n × (n - 1) × ... × 2 × 1`. By definition, `0! = 1`.

How many ways can you choose a 5-card poker hand from a 52-card deck?

Since card order in hand does not matter: `52C5 = 52! / (5! × 47!) = 2,598,960` possible combinations.

How many 4-digit PIN codes can be created?

With repetition allowed: `10^4 = 10,000` PINs. Without repetition: `10P4 = 10! / 6! = 5,040` permutations.

Why is nCr always less than or equal to nPr?

Because combinations divide permutations by `r!` (the number of ways to reorder the chosen items), reducing the count unless `r = 0` or `r = 1`.

What is the difference between combinations and permutations?

Permutations care about order (e.g., lock combinations, race standings), whereas Combinations ignore order (e.g., lottery numbers, hand of poker cards).

Math & Statistics Cluster

Combination & Permutation Calculator (nCr & nPr Formulas)

Calculate combinations (order does not matter) and permutations (order matters) using factorials.

TL;DR: Combination & Permutation Calculator computes $nCr = n! / [r!(n-r)!]$ and $nPr = n! / (n-r)!$ to determine total arrangement and selection counts.

n (total items)

r (items chosen)

What Is the Difference Between Combinations (nCr) and Permutations (nPr)?

The core difference is whether order matters. In permutations (nPr), order matters (e.g., race podium positions, passwords): nPr = n! / (n - r)!. In combinations (nCr), order does not matter (e.g., lottery numbers, committee members): nCr = n! / [r!(n - r)!].

How to Use the Combination & Permutation Calculator

Our Combination & Permutation Calculator performs high-precision mathematical operations directly in your browser with zero latency and complete client-side privacy.

Enter total number of items in the set ($n$).
Enter number of items to select ($r$), where $0 \le r \le n$.
Select whether Order Matters (Permutation) or Order Does Not Matter (Combination).
Optionally toggle 'Repetition Allowed'.
View calculated permutations, combinations, and step-by-step factorial expansions.

Mathematical Formula & Equations

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

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

Calculation Example

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

100% Client-Side Privacy & Data Security

All calculations, amortization schedules, variables, and sensitive numerical datasets execute 100% locally in your web browser memory. Your financial, medical, and personal values are never transmitted, logged, or uploaded to any external server.

Frequently Asked Questions

: Combinations: `nCr = n! / [r!(n - r)!]`. Permutations: `nPr = n! / (n - r)!`, where `!` represents the factorial operation.
: A factorial is the product of all positive integers less than or equal to n: `n! = n × (n - 1) × ... × 2 × 1`. By definition, `0! = 1`.
: Since card order in hand does not matter: `52C5 = 52! / (5! × 47!) = 2,598,960` possible combinations.
: With repetition allowed: `10^4 = 10,000` PINs. Without repetition: `10P4 = 10! / 6! = 5,040` permutations.
: Because combinations divide permutations by `r!` (the number of ways to reorder the chosen items), reducing the count unless `r = 0` or `r = 1`.
: Permutations care about order (e.g., lock combinations, race standings), whereas Combinations ignore order (e.g., lottery numbers, hand of poker cards).

Cite This Calculator

Need to reference this online calculator in a financial proposal, academic paper, blog post, or classroom curriculum? Copy a citation below.

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

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

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

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

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