Factorial Calculator
Result:
Calculate the factorial of any non-negative integer with our free online factorial calculator. Get instant results with step-by-step calculations and scientific notation for large values.
What is a Factorial?
The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. By definition, 0! = 1.
Factorial Formula
n! = n × (n-1) × (n-2) × ... × 2 × 1Special case: 0! = 1
Factorial Properties
Basic Properties
- 0! = 1 (by definition)
- 1! = 1
- n! = n × (n-1)!
- Recursive nature: Each factorial builds on the previous
- Growth rate: Extremely fast growth
Mathematical Uses
- Permutations: P(n,r) = n!/(n-r)!
- Combinations: C(n,r) = n!/(r!(n-r)!)
- Series expansions: Taylor series
- Probability: Arrangements and orderings
- Calculus: Integration and differentiation
Step-by-Step Factorial Examples
Problem: Calculate 5!
Step-by-step solution:
5! = 5 × 4 × 3 × 2 × 1
5! = 5 × 4! = 5 × 24 = 120
Result: 5! = 120
Special cases:
0! = 1 (by mathematical definition)
1! = 1
Why 0! = 1? This definition makes mathematical formulas work consistently, especially in combinatorics where C(n,0) = n!/0! = 1.
Problem: Calculate 10!
Method 1 - Direct calculation:
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
Method 2 - Using previous result:
10! = 10 × 9! = 10 × 362,880 = 3,628,800
Result: 10! = 3,628,800
Common Factorial Values
| n | n! | Growth Rate |
|---|---|---|
| 0 | 1 | - |
| 1 | 1 | ×1 |
| 2 | 2 | ×2 |
| 3 | 6 | ×3 |
| 4 | 24 | ×4 |
| 5 | 120 | ×5 |
| 6 | 720 | ×6 |
| 7 | 5,040 | ×7 |
| 8 | 40,320 | ×8 |
| 9 | 362,880 | ×9 |
| 10 | 3,628,800 | ×10 |
Applications of Factorials
Combinatorics
- Permutations and arrangements
- Combinations and selections
- Probability calculations
- Counting principles
- Game theory problems
Mathematical Analysis
- Taylor series expansions
- Stirling's approximation
- Gamma function extensions
- Power series convergence
- Calculus applications
Computer Science
- Algorithm complexity analysis
- Recursive programming
- Dynamic programming
- Big O notation
- Computational mathematics
Factorial Growth and Limits
⚠️ Extremely Fast Growth
Factorial growth is extremely rapid:
- 20! = 2.43 × 10¹⁸ (2.4 quintillion)
- 50! = 3.04 × 10⁶⁴ (3.04 vigintillion)
- 100! = 9.33 × 10¹⁵⁷ (requires scientific notation)
- 170! is approximately the largest factorial that can be represented in standard floating-point arithmetic
Stirling's Approximation: For large n, n! ≈ √(2πn) × (n/e)ⁿ
Factorial in Programming
Recursive Implementation
function factorial(n) {
if (n === 0 || n === 1) {
return 1;
}
return n * factorial(n - 1);
}Pros: Simple, intuitive
Cons: Stack overflow risk for large n
Iterative Implementation
function factorial(n) {
let result = 1;
for (let i = 2; i <= n; i++) {
result *= i;
}
return result;
}Pros: Memory efficient, no stack overflow
Cons: Less intuitive than recursive
Frequently Asked Questions (FAQ)
0! = 1 by mathematical definition. This might seem counterintuitive, but it's defined this way because:
- It makes mathematical formulas work consistently
- There's exactly one way to arrange zero objects (the empty arrangement)
- It ensures that n!/n! = 1 for all n ≥ 0
- Combinatorial formulas like C(n,0) = n!/0! = 1 work correctly
Our calculator can handle up to 170! Beyond this:
- 170! is approximately 7.26 × 10³⁰⁶
- Larger factorials require scientific notation or special mathematical libraries
- For very large factorials, use Stirling's approximation: n! ≈ √(2πn) × (n/e)ⁿ
- Programming languages may have different limits based on data types
Factorials have many practical applications:
- Permutations: Arranging people in a line, seating arrangements
- Combinations: Selecting team members, lottery calculations
- Probability: Card games, statistical analysis
- Cryptography: Security algorithms and key generation
- Physics: Statistical mechanics and quantum mechanics
- Economics: Portfolio theory and risk analysis
Both use factorials but serve different purposes:
Permutations (Order Matters)
- Formula: P(n,r) = n!/(n-r)!
- Example: Arranging 3 people in a line from 5 people
- Result: ABC is different from BAC
Combinations (Order Doesn't Matter)
- Formula: C(n,r) = n!/(r!(n-r)!)
- Example: Selecting 3 people from 5 people
- Result: ABC is the same as BAC
No, factorials are only defined for non-negative integers.
- Factorials of negative numbers are undefined in standard mathematics
- The factorial function can be extended using the Gamma function: Γ(n+1) = n!
- For non-integers, use the Gamma function: Γ(z) = ∫₀^∞ t^(z-1)e^(-t) dt
- Our calculator only accepts non-negative integers (0, 1, 2, 3, ...)
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