About

In statistical mathematics, the distinction between order and selection is the fundamental barrier to accuracy. This tool bridges that gap by resolving the two primary distinct functions of combinatorics: Combination (nCr) and Permutation (nPr).

Miscalculation in probability theory often stems from ignoring boundary conditions. Factorials scale super-exponentially; calculating outcomes for just n=100 exceeds the atoms in the universe. This engine utilizes BigInt arithmetic to process standard combinatorial explosion without floating-point errors, strictly handling values where n ≥ r ≥ 0.

Formulas

The calculator uses the optimized multiplicative formulas to reduce overflow risks during intermediate steps.

Permutation: r∏i=1 (n − i + 1)

Combination: 1r! r∏i=1 (n − i + 1)

Where n is the total set size and r is the subset size. The logic automatically cancels terms before division to maximize precision.

Reference Data

Concept	Notation	Formula	Order Matters?	Repetition?
Permutation	P(n,r)	n!(n − r)!	YES	FALSE
Combination	C(n,r)	n!r!(n − r)!	NO	FALSE
Repetitive Permutation	n^r	n^r	YES	TRUE
Multiset Coefficient	H(n,r)	(n + r − 1)!r!(n − 1)!	NO	TRUE

Frequently Asked Questions

Think of a safe vs. a salad. In a "Combination" (Salad), the order of ingredients (Lettuce, Tomato, Onion) doesn't matter; it's the same salad. In a "Permutation" (Safe Lock), the order (12-30-5) is critical; 30-12-5 will not open the safe. If order implies a different outcome, use nPr.

Combinatorics generate numbers that grow incredibly fast. 52 factorial (the number of ways to shuffle a deck of cards) has 68 digits. To maintain a responsive UI and memory safety, results exceeding 30 digits are formatted scientifically, though internal calculations retain BigInt precision where possible.

By definition in mathematics, 0! = 1. This is an empty product convention allowing combinatorial formulas to work consistently. Our engine validates n and r such that if r=0 or n=r, the result returns 1 immediately.

Geometric expansion. Generating a list of all combinations is a O(2^n) complexity task. While calculating the count is instant, rendering 20 items (2^20) would generate over 1,000,000 visual elements, which would freeze your browser. We cap the visual generator at N=12 (4096 items) for performance.