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Factorial Calculator (n!) with Expressions & Permutations

Step-by-Step • Expressions • Permutations & Combinations

Factorial (n!)
Calculatefactorial
120
5! = 5 × 4 × 3 × 2 × 1
= 5 × 4!
= 20 × 3!
= 60 × 2!
= 120 × 1
= 120
Permutation (nPr)
Arrangefromitems
10P3 = 720
Order matters: nPr = n!/(n-r)!
Combination (nCr)
Choosefromitems
10C3 = 120
Order doesn't matter: nCr = n!/(r!(n-r)!)
Expression Solver
Supports: +, -, *, /, ^, %, ()

Popular Examples

5 factorial
= 120
Arrange 5 items
10 choose 3
= 120
Select 3 from 10
10 permute 3
= 720
Arrange 3 from 10
52!
= 8.07×10⁶⁷
Shuffle deck of cards
100!/98!
= 9,900
Simplify expression
0 factorial
= 1
Zero factorial equals 1
20!
= 2.43×10¹⁸
Large factorial
(6!+4!)/(5!)
= 6.2
Complex expression
7 choose 2
= 21
Lottery combinations
8 permute 3
= 336
Podium positions
13!
= 6,227,020,800
Calendar permutations
170!
= 7.26×10³⁰⁶
Maximum factorial

Recent Calculations

10C3→120
10P3→720
5!→120

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.

How to Use

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

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

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

Tips & Best Practices

💡

Use the expression solver for ratios like 10!/5! to avoid overflow

💡

Check the step-by-step breakdown to understand the calculation

💡

Use permutations (nPr) when order matters (e.g., race positions)

💡

Use combinations (nCr) when order doesn't matter (e.g., lottery)

💡

Remember: 0! = 1 by mathematical definition

💡

Maximum factorial is 170! due to computational limits

💡

Use scientific notation for very large results

💡

Save time by using popular examples for common calculations

Related Tools

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Understanding Factorials

What is a Factorial?

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
  • 0! = 1 (by definition)
  • 1! = 1

Permutations vs Combinations

Permutations (nPr)

Order matters

Formula: nPr = n!/(n-r)!

Example: Arranging 3 people from 10 for podium positions (1st, 2nd, 3rd)

10P3 = 10!/7! = 720 ways

Combinations (nCr)

Order doesn't matter

Formula: nCr = n!/(r!(n-r)!)

Example: Choosing 3 numbers from 10 for lottery

10C3 = 10!/(3!×7!) = 120 ways

Factorial Table (Quick Reference)

nn!Value
11!1
22!2
33!6
44!24
55!120
66!720
77!5040
88!40320
99!362880
1010!3628800
1111!3.99168 E+7
1212!4.790016 E+8
1313!6.2270208 E+9
1414!8.71782912 E+10
1515!1.307674368 E+12
2020!2.432902008 E+18
2525!1.551121004 E+25
3030!2.652528598 E+32
5050!3.04140932 E+64
100100!9.332621544 E+157
170170!7.257415615 E+306

Frequently Asked Questions

What is a factorial?

A factorial (n!) is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in combinatorics, probability, and algebra.

How do you calculate factorial?

To calculate n!, multiply all integers from n down to 1. For example: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. Our calculator shows step-by-step breakdowns for easy understanding.

What is 0 factorial (0!)?

0! equals 1 by mathematical definition. This is because n! = n × (n-1)!, so 1! = 1 × 0!, which means 0! = 1!/1 = 1. This definition is essential for combinatorics formulas.

What is the difference between permutation and combination?

Permutations (nPr) count arrangements where order matters (e.g., race positions: 1st, 2nd, 3rd). Combinations (nCr) count selections where order doesn't matter (e.g., lottery numbers). Formula: nPr = n!/(n-r)! and nCr = n!/(r!(n-r)!).