combination

Calculates the number of combinations (n choose r) using efficient iterative method. Computes the binomial coefficient C(n, r) = n! / (r! * (n-r)!) using an optimized algorithm that avoids computing large factorials. Automatically uses the symmetry property C(n, r) = C(n, n-r) to minimize iterations and prevent overflow.

Signature

const combination: (n: number, r: number) => number

Parameters

Name	Type	Description
n	-	Total number of items (must be non-negative integer)
r	-	Number of items to choose (must be non-negative integer)

Returns

Number of ways to choose r items from n items

Examples

Basic combination calculations

import { combination } from '@winglet/common-utils';

console.log(combination(5, 2)); // 10 - choosing 2 items from 5
console.log(combination(6, 4)); // 15 - choosing 4 items from 6
console.log(combination(10, 5)); // 252 - choosing 5 items from 10

// Edge cases
console.log(combination(5, 0)); // 1 - choosing 0 items (empty set)
console.log(combination(5, 5)); // 1 - choosing all items
console.log(combination(5, 6)); // 0 - impossible to choose more than available

Symmetry property and optimization

// Uses symmetry C(n, r) = C(n, n-r) for efficiency
console.log(combination(20, 3)); // 1140 - optimized to compute C(20, 3)
console.log(combination(20, 17)); // 1140 - optimized to compute C(20, 3)

// Large combinations that would overflow with factorial approach
console.log(combination(50, 25)); // 126410606437752
console.log(combination(100, 2)); // 4950

Playground

import { combination } from '@winglet/common-utils';

console.log(combination(5, 2)); // 10 - choosing 2 items from 5
console.log(combination(6, 4)); // 15 - choosing 4 items from 6
console.log(combination(10, 5)); // 252 - choosing 5 items from 10

// Edge cases
console.log(combination(5, 0)); // 1 - choosing 0 items (empty set)
console.log(combination(5, 5)); // 1 - choosing all items
console.log(combination(5, 6)); // 0 - impossible to choose more than available

Notes

Mathematical Properties:

• Returns 1 when r = 0 or r = n (choosing nothing or everything)
• Returns 0 when r > n (impossible combinations)
• Uses symmetry C(n, r) = C(n, n-r) to optimize performance
• Avoids factorial computation to prevent integer overflow

Use Cases:

• Probability calculations and combinatorial analysis
• Generating lottery odds and gambling probabilities
• Algorithm complexity analysis (choosing subsets)
• Statistical sampling and experimental design
• Graph theory (selecting vertices or edges)
• Binomial expansion coefficients

Performance: O(min(r, n-r)) time complexity, O(1) space complexity. Much more efficient than factorial-based approaches for large values.