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Pure & Applied MathematicsEntry № 03.08Documented Formula

Permutation & Combination Calculator (nPr & nCr)

Computes permutations nPr (order matters) and combinations nCr (order does not matter) with full factorial factor expansions.

Popular Searches:Formula for nCr: n! / (r!(n-r)!)Difference between permutation and combination with examplesHow many combinations of 5 from 52 cards?
Combinatorics Engine: Permutation (nPr) & Combination (nCr)
Order Matters (nPr) vs Order Free (nCr)
n = 8
r = 3
Combinations ⁿCᵣ (Unordered Selections)
56
Formula: n! / [r!(n−r)!]
Permutations ⁿPᵣ (Ordered Arrangements)
336
Formula: n! / (n−r)!
Mathematical Proof & FormulaStandard Mathematical Notation

How This Calculation Formula is Formulated

Evaluates combinatorial selections and arrangements. Permutations (nPr) account for ordering, whereas Combinations (nCr) count unordered subset selections from n distinct elements.

Standard Algebraic Representation
nPr = n! / (n − r)! ; nCr = n! / [r! · (n − r)!]
Documented algebraic formula with transparent derivation and reference notes.

Variable Definitions & Measurement Units

n
Total Elements
Total population size.
r
Sample Size
Number of elements selected or ordered.
Step-by-Step Calculation Example

Selecting and Arranging 3 Items from 8

Computing combinations and permutations for n = 8, r = 3.

1
Factorials
8! = 40,320 ; 5! = 120 ; 3! = 6
Factorial computations
2
Permutations (nPr)
8! / 5! = 40,320 / 120 = 336
Order matters (e.g. 1st, 2nd, 3rd place)
3
Combinations (nCr)
8! / (3! × 5!) = 40,320 / (6 × 120) = 56
Order does not matter (e.g. committee)
Conclusion: There are 56 unique combinations (unordered groups) and 336 permutations (ordered sequences).
In-Depth Editorial Analysis

Pascal's Triangle and Binomial Coefficients

Combinations (nCr) are identical to the binomial coefficients in the expansion of (x + y)ⁿ and form the entries of Pascal's Triangle.

Symmetry identity states that nCr = nC(n-r). For instance, choosing 3 items out of 8 is identical to choosing the 5 items that are left behind (8C3 = 8C5 = 56).

Frequently Asked Questions

Frequently Asked Questions About Permutation & Combination Calculator (nPr & nCr)

How do you remember permutation vs combination?

Permutation = Position (order matters). Combination = Committee (order does not matter).

What is 0! (zero factorial)?

By mathematical definition and convention, 0! = 1, ensuring formulas like nCn = n! / (n! · 0!) = 1 hold true.

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Deterministic Precision & 100% Client-Side Privacy

All calculations execute in your local browser using IEEE 754 double-precision floating-point arithmetic. Your figures and financial metrics remain private and are never uploaded or saved to any cloud servers.