Math July 13, 2026 · 8 Min Read

Permutation & Combination Calculator – Guide & Formulas

Calculate standard permutations and combinations with or without repetitions using step-by-step factorial formulas.

The Permutation and Combination Calculator resolves core combinatorics problems. It computes standard variations (nPr, nCr) and repetitive models with rigorous mathematical steps.

Key Takeaway

Use the free Permutation & Combination Calculator to calculate standard permutations and combinations with or without repetitions using step-by-step factorial formulas. Get instant results with step-by-step explanations.

How to Use the Permutation & Combination Calculator

  1. Enter the total items count (n) up to 170.
  2. Enter the selected items count (r) up to 170.
  3. Ensure inputs are non-negative integers.
  4. Review C(n,r) and P(n,r) values alongside repetition equivalents.

The Formula

P(n, r) = n! / (n-r)! ; C(n, r) = n! / (r! * (n-r)!)

Variable Definitions

  • n!: Factorial representing product of all positive integers up to n
  • r: The subset of elements chosen from the collection

Selecting 3 items from a pool of 10

Calculate permutations and combinations for n = 10 and r = 3.

  1. Step 1: P(10, 3) = 10! / (10-3)! = 10! / 7! = 10 * 9 * 8 = 720.
  2. Step 2: C(10, 3) = P(10, 3) / 3! = 720 / 6 = 120.
  3. Step 3: Permutations with repetition n^r = 10^3 = 1,000.
  4. Step 4: Combinations with repetition (10+3-1)! / (3! * 9!) = 12! / (6 * 9!) = 220.

Frequently Asked Questions

What is the main difference between permutations and combinations?

In Permutations, order matters (e.g. locks, codes). In Combinations, order does not matter (e.g. card hands, team selections).

Why is n limited to 170?

Because factorials grow exponentially; 171! exceeds the maximum double-precision floating point number capacity (approx. 1.8 × 10³⁰⁸) in computer architectures.