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Factorial Calculator (n!) with Expressions & Permutations

Step-by-Step Breakdown • Expressions Solver • Permutations & Combinations

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Factorial & Expression Solver

Calculate exact factorials, solve equations (e.g. 5! + 3! or 10!/5!), or use the slider

5!
Type an integer (e.g. 10), a factorial notation (10!), or complex formulas.
5
Drag to instantly compute exact factorials from 0! to 170!.
Result (5!):
120

Step-by-Step Multiplication

5! = 5 × 4 × 3 × 2 × 1
= 5 × 4!
= 20 × 3!
= 60 × 2!
= 120 × 1
= 120

Mathematical Presets & Benchmarks

Quickly load specific mathematical constants, arrangements, and combinatorics limits

Arrangements (nPr) & Selections (nCr)

Calculate probabilities, permutations, and choices using optimized overflow-proof algorithms

Permutations (nPr)
n! / (n - r)!
10P3 =720
Order matters (e.g. race finishes, passcode digits).
Combinations (nCr)
n! / [r! × (n - r)!]
10C3 =120
Order does not matter (e.g. lottery picks, team formations).

Stirling's Approximation Studio

Estimate giant factorials up to 10,000! and track mathematical error tolerances

Stirling's theorem calculates $\ln(n!) \approx n\ln(n) - n + 0.5\ln(2\pi n)$ to bypass double-precision overflow limits.

Error Matrix (Exact vs Stirling)

Target Factorial:5!
Exact Value:120
Stirling's Estimate:1.180 × 10^2
Absolute Error:1.98
Relative Margin:1.6500%
💡 Margin drops below **1%** at $n \ge 10$. As $n \to \infty$, relative error approaches zero.

Recent Tape Calculations

combination 10C3→120
permutation 10P3→720
factorial 5!→120

Factorial Reference Chart (n! Quick Guide)

Instant numerical guides for small and landmark factorial inputs

Input (n)Factorial Notation (n!)Exact Value / Decimal Form
11!1
22!2
33!6
44!24
55!120
66!720
77!5,040
88!40,320
99!362,880
1010!3,628,800
1111!3.992 × 10^7
1212!4.790 × 10^8
1313!6.227 × 10^9
1414!8.718 × 10^10
1515!1.308 × 10^12
2020!2.433 × 10^18
2525!1.551 × 10^25
3030!2.653 × 10^32
5050!3.041 × 10^64
100100!9.333 × 10^157
170170!7.257 × 10^306

Overview & Capabilities

Calculate factorials (n!), permutations (nPr), and combinations (nCr) with our advanced calculator. Features step-by-step breakdowns, expression parsing, visual growth charts, and calculation history for easy understanding of factorial mathematics.

Tutorial

How to Use

01
Enter a number (0-170) to calculate its factorial
02
Use the expression solver for complex calculations like 10!/5! or (6!+4!)/(5!)
03
Calculate permutations (nPr) to find arrangements of r items from n items
04
Calculate combinations (nCr) to find selections of r items from n items
05
View step-by-step breakdowns to understand the calculation process
06
Copy or share your results with one click
Capabilities

Key Features

Factorial Calculation: Compute n! for numbers 0 to 170 instantly
Expression Parser: Solve complex expressions with multiple factorials
Permutation Calculator: Calculate nPr = n!/(n-r)! for arrangements
Combination Calculator: Calculate nCr = n!/(r!(n-r)!) for selections
Step-by-Step Breakdown: Visual expansion showing each multiplication
Power Search: Natural language input like "10 factorial" or "10 choose 3"
Calculation History: Review and reload your last 10 calculations
Popular Examples: Real-world scenarios like deck shuffling, lottery odds
Scientific Notation: Automatic formatting for very large numbers
Mobile Optimized: Touch-friendly inputs and responsive design
Applications

Common Use Cases

Mathematics Education: Learn factorial concepts with visual breakdowns
Combinatorics: Calculate permutations and combinations for probability
Algorithm Analysis: Understand O(n!) time complexity
Statistics: Compute binomial coefficients and probability distributions
Programming: Verify factorial implementations and edge cases
Puzzle Solving: Calculate possible arrangements and selections
Game Theory: Determine possible game states and outcomes
Cryptography: Understand keyspace sizes and combinations
Guidance

Tips & Best Practices

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Use the expression solver for ratios like 10!/5! to avoid overflow
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Check the step-by-step breakdown to understand the calculation
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Use permutations (nPr) when order matters (e.g., race positions)
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Use combinations (nCr) when order doesn't matter (e.g., lottery)
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Remember: 0! = 1 by mathematical definition
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Maximum factorial is 170! due to computational limits
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Use scientific notation for very large results
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Save time by using popular examples for common calculations
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Answers

Frequently Asked Questions

Q What is a factorial, and how is n! calculated?

A factorial (denoted as n!) is the product of all positive integers less than or equal to n. For example: 5! = 5 × 4 × 3 × 2 × 1 = 120. It represents the number of ways to arrange n distinct objects.

Q Why does 0 factorial (0!) equal 1?

By mathematical definition, 0! = 1. This is consistent with combinatorics formulas (there is exactly 1 way to arrange 0 items: doing nothing) and the recursive relationship n! = n × (n-1)!, which for n=1 yields 1! = 1 × 0!, meaning 0! must equal 1.

Q What is the difference between permutations (nPr) and combinations (nCr)?

Permutations (nPr) count arrangements where the order matters (e.g. race podium finishes: 1st, 2nd, 3rd). Combinations (nCr) count selections where the order does not matter (e.g. choosing 5 lottery numbers). nPr = n! / (n-r)! and nCr = n! / (r! × (n-r)!).

Q What is Stirling's Approximation and when is it used?

Stirling's Approximation is an analytical formula used to estimate factorials: n! ≈ sqrt(2 × π × n) × (n / e)^n. It is extremely useful in physics, probability, and computer science for calculating huge factorials where exact multiplication is computationally expensive or overflows standard numerical types.

Q Why does a standard calculator throw an error for factorials above 170?

Computers use the IEEE 754 double-precision floating-point format to store large decimal values. This format has a maximum value limit of approximately 1.79 × 10^308. Since 170! is about 7.26 × 10^306 and 171! is 1.24 × 10^309, any factorial higher than 170 overflows this hardware limit and is represented as Infinity.