Permutation & Combination Calculator

Mastering Permutations and Combinations: Complete Guide

Permutations and combinations are the cornerstones of combinatorics, the branch of mathematics that studies counting. This Professional Permutation & Combination Calculator instantly computes nPr, nCr, and factorials, with optional repetition. It provides detailed step‑by‑step breakdowns, making it ideal for students, teachers, and anyone working with probability, statistics, or discrete mathematics.

Permutations (Order Matters)

A permutation counts the number of ways to arrange r items chosen from a set of n distinct items. The formula is P(n,r) = n! / (n−r)!. For example, the number of ways to arrange 3 books from a shelf of 10 is P(10,3) = 720. If repetition is allowed (e.g., a PIN code), the formula becomes n^r.

Combinations (Order Does Not Matter)

A combination counts the number of ways to select r items from a set of n without regard to order. The formula is C(n,r) = n! / (r!·(n−r)!). This is the binomial coefficient, often read as "n choose r". For example, the number of 5‑card hands from a 52‑card deck is C(52,5) = 2,598,960. With repetition allowed, the formula becomes C(n+r−1, r).

Factorials and Their Growth

The factorial function, denoted n!, is the product of all positive integers up to n. It grows extremely fast—10! is over 3.6 million, and 20! is about 2.4×10¹⁸. This calculator handles factorials up to 170! (≈7.26×10³⁰⁶), beyond which JavaScript's double‑precision limit is reached.

Real‑World Applications

Combinatorics is essential in probability (lottery odds, poker hands), cryptography (key space sizes), genetics (allele combinations), and computer science (algorithm complexity). Understanding permutations and combinations empowers better decision‑making in fields ranging from business to gaming.