Permutation and Combination Calculator

Discrete Mathematics & Combinatorial Analysis

In probability theory, statistics, and counting mathematics, determining the total number of possible arrangements or subset selections of r objects drawn from a larger set of n distinct items depends on two essential criteria: order sensitivity and repetition allowance.

Core Combinatorial Formulas

• Permutations (Order Matters, No Replacement): nPr = P(n,r) = (n!) / ((n-r)!)

• Combinations (Order Irrelevant, No Replacement): nCr = C(n,r) = (n!) / (r!(n-r)!) = (nPr) / (r!)

• Permutations with Replacement: P_rep = n^r

• Combinations with Replacement: C_rep = n+r-1r = (!) / (r!(n-1)!)

Order vs Selection Decision Framework

• Order Matters (Ordered Sequences): Security PIN passcodes, athletic race podium finishes, lock combinations (which are technically mathematical permutations!).

• Order Irrelevant (Unordered Subsets): Poker hand card combinations, lottery ticket numbers, committee member selections.

Pascal's Triangle Symmetry Properties

Combinations exhibit mathematical symmetry such that nCr = nC(n-r). For instance, selecting 3 items from 10 items yields 120 combinations, which identically equals leaving behind 7 items from

• This property simplifies large combinatorial computations in binomial theorem expansions.

How to Use This Calculator

Enter your Total Number of Items (n) (e.g. 10 in items) and Sample Size / Items Chosen (r) (e.g. 3 in chosen). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Formula Reference: Permutation and Combination Calculator

Input variables:

totalItemsN — Total Number of Items (n)

sampleSizeR — Sample Size / Items Chosen (r)

Computed outputs:

combinationsNCr — Combinations nCr (Order Irrelevant)

permutationsNPr — Permutations nPr (Order Matters)

combinationsWithReplacement — Combinations with Replacement

permutationsWithReplacement — Permutations with Replacement (n^r)

Mathematical relationships extracted from calculation logic:

nPr = safeDiv(factorial(n), factorial(n - r))

nCrRep = safeDiv(factorial(n + r - 1), factorial(r) * factorial(n - 1))

Standard: NIST / ISO mathematical definitions

Worked Example

Choosing 3 winners out of 10 contestants: If order matters (1st, 2nd, 3rd place prizes), there are 10P3 = 720 permutations. If order does not matter (3 identical prizes), there are 10C3 = 120 combinations.

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.