Factorial Calculator
Calculate the factorial of any number and understand factorial concepts
Factorial Calculation Results
Factorial Result
Scientific Notation
Number of Digits
Common Factorial Values
| Number (n) | Factorial (n!) | Applications |
|---|---|---|
| 0 | 1 | Empty product convention |
| 1 | 1 | Single arrangement |
| 5 | 120 | Small permutations |
| 10 | 3,628,800 | Medium permutations |
| 52 | 8.0658 × 10⁶⁷ | Card deck arrangements |
Factorial Calculation Tips
- Factorials grow extremely fast – 10! is already 3.6 million
- 0! is defined as 1 by mathematical convention
- Factorials are only defined for non-negative integers
- For large numbers, factorials are expressed in scientific notation
- Factorials are used in permutations, combinations, and probability
Understanding Factorials
The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. Factorials are fundamental in mathematics, particularly in combinatorics, algebra, and mathematical analysis.
For example, 5! (read as “five factorial”) is calculated as 5 × 4 × 3 × 2 × 1 = 120. Factorials grow at an extremely rapid rate, which makes them useful for counting permutations and combinations in probability theory.
Applications of Factorials
Combinatorics
Counting permutations and combinations of objects
Probability
Calculating probabilities in various scenarios
Series Expansions
Used in Taylor series and other mathematical expansions
Algorithms
Important in computer science for recursive algorithms
Key Properties of Factorials
Factorials have several important mathematical properties:
- Recursive Definition: n! = n × (n-1)! with 0! = 1
- Rapid Growth: Factorials grow faster than exponential functions
- Gamma Function: Factorials can be extended to complex numbers using the gamma function
- Stirling’s Approximation: n! ≈ √(2πn) × (n/e)ⁿ for large n
- Divisibility: n! is divisible by all integers from 1 to n
How to Use Our Factorial Calculator
Enter Your Number
Input a non-negative integer between 0 and 170
Click Calculate
Press the calculate button to compute the factorial
View Results
See the factorial result, scientific notation, and digit count
Explore Applications
Learn about how factorials are used in mathematics
Real-World Examples of Factorials
Factorials appear in many practical situations:
Seating Arrangements
The number of ways to arrange n people in a row is n!
Tournament Scheduling
Factorials help calculate possible match arrangements
Genetic Sequences
Counting possible arrangements of genetic elements
Password Combinations
Calculating possible permutations for security
Frequently Asked Questions
0! is defined as 1 by convention for several reasons: it maintains consistency in mathematical formulas, it represents the number of ways to arrange zero objects (which is 1 way – the empty arrangement), and it makes recursive formulas work properly.
This calculator can handle factorials up to 170! due to JavaScript’s number precision limitations. Beyond this point, numbers become too large to represent accurately as standard JavaScript numbers. For larger factorials, specialized mathematical software is needed.
Standard factorials are only defined for non-negative integers. However, the gamma function extends the concept of factorials to complex numbers. The gamma function Γ(z) is defined for all complex numbers except negative integers, and Γ(n+1) = n! for positive integers n.
Factorials are fundamental in probability theory, particularly in permutations and combinations. The number of ways to arrange n distinct objects is n! (permutations). The binomial coefficient, used in combinations, is calculated as n!/(k!(n-k)!), which gives the number of ways to choose k items from n without regard to order.
Stirling’s approximation is a formula for approximating factorials of large numbers: n! ≈ √(2πn) × (n/e)ⁿ. This approximation becomes increasingly accurate as n grows larger and is useful when calculating factorials of very large numbers where exact computation is impractical.