Factorial Calculator
Calculate factorials (n!) instantly with our advanced calculator. Get detailed step-by-step solutions and understand the mathematical process behind factorial calculations.
Factorial Tool
Enter a number and get its factorial with detailed step-by-step solution
Master Factorial Calculations with Our Advanced Calculator
Our factorial calculator is an essential tool for students, mathematicians, and anyone working withcombinatorics, probability, and mathematical analysis. Whether you're solvingmath homework, working with permutations and combinations, studying statistics, or exploring number theory, this tool provides comprehensive solutions with step-by-step explanations.
The factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . Our factorial calculator uses the recursive definition with the base case to efficiently compute factorials. This math calculator is particularly useful for combinatorics problems, probability calculations, and mathematical analysis where factorials frequently appear.
Perfect for high school students learning about permutations and combinations, college studentsstudying probability and statistics, mathematics majors exploring advanced topics, andprofessionals in data science and research. The factorial calculator toolprovides not just the final result, but also the complete mathematical process showing how each step contributes to the final answer.
Factorial Applications and Properties
| Application | Formula | Description | Example |
|---|---|---|---|
| Permutations | P(n,r) = n!/(n-r)! | Number of ways to arrange r objects from n | P(5,3) = 5!/(5-3)! = 60 |
| Combinations | C(n,r) = n!/(r!(n-r)!) | Number of ways to choose r objects from n | C(5,3) = 5!/(3!2!) = 10 |
| Taylor Series | e^x = Σ(x^n/n!) | Exponential function expansion | e = 1 + 1/1! + 1/2! + 1/3! + ... |
| Stirling Numbers | S(n,k) = k! × S(n,k) | Partitioning n objects into k subsets | S(4,2) = 7 |
| Gamma Function | Γ(n) = (n-1)! | Extension of factorial to real numbers | Γ(4) = 3! = 6 |
Common Mistakes to Avoid
Negative Numbers
Factorial is only defined for non-negative integers. For negative numbers, factorial is undefined. The gamma function extends factorial to real numbers, but is undefined for positive integers .
Confusing with Exponentiation
is not the same as . For example, , but .
Forgetting 0! = 1
By definition, . This is necessary for many combinatorial formulas to work correctly, such as and .
How to Calculate Factorial
Factorial calculation is straightforward but can become computationally intensive for large numbers. The recursive definition provides a clear mathematical foundation for understanding factorials.
This recursive formula allows us to calculate factorials efficiently. The base case is , and we can build up to any positive integer by multiplying by the factorial of .
Factorial Calculation Steps
Check input
Ensure n is a non-negative integer
Base case
If n = 0, return 1
Recursive case
Calculate n × (n-1)!
Iterative approach
Multiply from 1 to n
Result
Return the final product
Key Properties of Factorial
Growth Rate
Factorial grows faster than exponential: for
Stirling Approximation
For large :
Examples
Small Number
Number: 5
Calculation:
- 5! = 5 × 4 × 3 × 2 × 1
- 5! = 5 × 24
- 5! = 120
Medium Number
Number: 10
Calculation:
- 10! = 10 × 9 × 8 × ... × 1
- 10! = 10 × 362,880
- 10! = 3,628,800
Zero Factorial
Number: 0
Calculation:
- By definition: 0! = 1
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Solve with AskMathAIFrequently Asked Questions
What is a factorial?
A factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.
Why is 0! = 1?
0! = 1 by definition. This is necessary for many mathematical formulas to work correctly, such as combinations C(n,0) = 1 and C(n,n) = 1. It also makes the recursive formula n! = n × (n-1)! work for n = 1.
What are the applications of factorial?
Factorials are used in combinatorics (permutations and combinations), probability theory, Taylor series expansions, statistical analysis, and many areas of mathematics and computer science.
How fast does factorial grow?
Factorial grows very rapidly. For example, 10! = 3,628,800, 20! ≈ 2.4 × 10^18, and 50! ≈ 3.0 × 10^64. Factorial grows faster than exponential functions.
Can factorial be calculated for negative numbers?
No, factorial is only defined for non-negative integers. For negative numbers, factorial is undefined. However, the gamma function extends factorial to real numbers (except negative integers).
What is the relationship between factorial and combinations?
Combinations use factorial in their formula: C(n,r) = n!/(r!(n-r)!). This gives the number of ways to choose r objects from n objects without considering order.