Factorial Calculator
Calculate factorials with step-by-step solutions and mathematical insights
n! = n × (n-1) × (n-2) × ... × 2 × 1
The factorial of n is the product of all positive integers less than or equal to n
Must be a non-negative integer (0 to 170)
Quick Examples
Invalid Input
! (factorial)
digits
Regular Factorial
!
Double Factorial
!!
Factorial Table: 0! to !
| n | n! |
|---|---|
Table limited to 20! for display. Your requested ! =
Step-by-Step Breakdown
Regular Factorial
Double Factorial
Number of Digits
Stirling's Approximation
Common Factorial Values
| n | n! | Application |
|---|---|---|
| 3 | 6 | Ways to arrange 3 items |
| 5 | 120 | Permutations of 5 people |
| 7 | 5,040 | Days of the week arrangements |
| 10 | 3,628,800 | Bowling pin arrangements |
| 13 | 6,227,020,800 | Card deck arrangements (13 cards) |
| 52 | 8.07 × 10⁶⁷ | Full deck arrangements |
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About Factorial Calculator
What is a Factorial?
A factorial is the product of all positive integers from 1 to n, written as n! (pronounced "n factorial"). It's a fundamental concept in mathematics, particularly in combinatorics, probability, and calculus.
Factorial Formula
n! = n × (n-1) × (n-2) × ... × 2 × 1
Special cases:
- 0! = 1 (by definition)
- 1! = 1
How to Calculate Factorials
Example: 5!
5! = 5 × 4 × 3 × 2 × 1 = 120
Step-by-Step:
- Start with n
- Multiply by (n-1)
- Continue multiplying by decreasing integers
- Stop at 1
Common Factorial Values
| n | n! |
|---|---|
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 6 |
| 4 | 24 |
| 5 | 120 |
| 6 | 720 |
| 7 | 5,040 |
| 8 | 40,320 |
| 9 | 362,880 |
| 10 | 3,628,800 |
Double Factorial (n!!)
The double factorial multiplies every other number:
- Odd n: n!! = n × (n-2) × (n-4) × ... × 3 × 1
- Even n: n!! = n × (n-2) × (n-4) × ... × 4 × 2
Examples:
- 7!! = 7 × 5 × 3 × 1 = 105
- 8!! = 8 × 6 × 4 × 2 = 384
Key Properties
Recursive Definition
n! = n × (n-1)!
Gamma Function Relationship
For positive integers: n! = Γ(n+1)
Stirling's Approximation
For large n: n! ≈ √(2πn) × (n/e)^n
Applications
- Combinatorics - Counting permutations and combinations
- Probability - Calculating probabilities in statistics
- Taylor Series - Power series expansions
- Computer Science - Algorithm analysis
- Physics - Quantum mechanics and statistical mechanics
Factorial Growth
Factorials grow extremely fast! By 20!, the value exceeds 2.4 quintillion (2.4 × 10^18). This rapid growth is why factorials are important in complexity analysis.
Tips
- Factorials are only defined for non-negative integers
- Use the gamma function for non-integer "factorials"
- For very large n, use Stirling's approximation
- Remember: 0! = 1 (not 0!)