Combinatorics Calculator

Calculate permutations, combinations, and factorials with support for very large numbers using arbitrary-precision arithmetic.

What is Combinatorics?

Combinatorics is the branch of mathematics dealing with counting, arrangement, and combination of objects. The main concepts include:

Permutation (nPr)

A permutation is an arrangement of objects where order matters. The number of ways to arrange r items from n total items is:

ⁿP_r = n! / (n - r)!

For example, arranging 3 letters from the set {A, B, C, D} gives 4P3 = 24 arrangements (ABC, ABD, ACB, ACD, ...).

Combination (nCr)

A combination is a selection of objects where order does not matter. The number of ways to choose r items from n total items is:

ⁿC_r = n! / (r! × (n - r)!)

For example, choosing 3 letters from {A, B, C, D} gives 4C3 = 4 combinations ({A,B,C}, {A,B,D}, {A,C,D}, {B,C,D}).

Factorial (n!)

The factorial of n is the product of all positive integers less than or equal to n:

n! = n × (n - 1) × (n - 2) × ... × 2 × 1

Factorials grow very rapidly. For example, 20! = 2,432,902,008,176,640,000.

Applications of Combinatorics

Combinatorics has wide applications across many fields:

• Probability & Statistics: Calculating odds and sample spaces
• Computer Science: Algorithm analysis and data structures
• Cryptography: Key generation and encryption schemes
• Game Theory: Analyzing strategies and outcomes
• Biology: Genetic combinations and molecular structures
• Operations Research: Optimization and scheduling problems