Free Factorial Calculator - Calculate n! with Steps

Calculate factorials (n!) quickly and accurately with our free online factorial calculator. A factorial is the product of all positive integers from 1 to n, essential in mathematics, statistics, and combinatorics for calculating permutations, combinations, and probability.

Our calculator supports numbers up to 170! and provides step-by-step solutions, making it perfect for students learning about factorials and professionals working with mathematical computations. Get instant results with detailed explanations.

Calculate Factorial

!

Enter any non-negative integer from 0 to 170

Understanding Factorials: Complete Guide

Mathematical Definition

The factorial function is defined recursively:

n! = n × (n-1)!

0! = 1 (base case)

This recursive definition means that to calculate n!, you multiply n by the factorial of (n-1), continuing until you reach the base case of 0! = 1.

Growth Rate

Factorials grow extremely rapidly:

10! = 3,628,800

15! = 1,307,674,368,000

20! = 2.43 × 10¹⁸

100! ≈ 9.33 × 10¹⁵⁷

This rapid growth is why our calculator is limited to 170!, as larger values exceed standard number representations.

Real-World Applications

Cryptography

Used in RSA encryption and other cryptographic algorithms for generating secure keys and digital signatures.

Computer Science

Algorithm analysis, particularly in calculating time complexity for recursive algorithms and sorting methods.

Game Theory

Calculating possible outcomes in games, tournament arrangements, and strategic decision-making scenarios.

Factorial Formulas and Examples

Permutation Formula

P(n,r) = n!/(n-r)!

Number of ways to arrange r objects from n total objects where order matters.

Example: How many ways can you arrange 3 books from 5 books?

P(5,3) = 5!/(5-3)! = 5!/2! = 120/2 = 60 ways

Combination Formula

C(n,r) = n!/(r!(n-r)!)

Number of ways to choose r objects from n total objects where order doesnn't matter.

Example: How many ways can you choose 3 students from 10?

C(10,3) = 10!/(3!×7!) = 3,628,800/(6×5,040) = 120 ways

Stirlingn's Approximation

For very large values of n, Stirlingn's approximation provides an estimate:

n! ≈ √(2πn) × (n/e)ⁿ

This approximation becomes more accurate as n increases and is useful in statistical mechanics and probability theory.

Tips for Working with Factorials

Computational Efficiency

For large calculations, use the recursive property: n! = n × (n-1)!. This allows you to build up from smaller factorials and avoid recalculating common terms.

Cancellation in Fractions

When working with ratios like P(n,r) = n!/(n-r)!, many terms cancel out. Instead of calculating full factorials, compute n×(n-1)×...×(n-r+1) directly.

Memory Techniques

Remember key values: 0!=1, 1!=1, 2!=2, 3!=6, 4!=24, 5!=120. These building blocks help with mental calculations and verification of larger results.

Understanding Zero Factorial

0! = 1 might seem counterintuitive, but itn's defined this way to make formulas work consistently. It represents the number of ways to arrange zero objects (exactly one way: do nothing).

Large Number Handling

For factorials beyond 170!, consider using logarithms: log(n!) = log(1) + log(2) + ... + log(n). This prevents overflow and allows comparison of very large factorials.

Problem-Solving Strategy

When encountering factorial problems, identify if you need permutations (order matters) or combinations (order doesnn't matter). This determines whether to use P(n,r) or C(n,r) formulas.