Factorial Calculator
Results:
Understanding Factorials and Permutations
What is a Factorial?
A factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n.
n! = n × (n-1) × (n-2) × ... × 2 × 1
Special cases:
0! = 1
1! = 1
Properties and Applications
Key Properties
- n! = n × (n-1)!
- (n+1)! = (n+1) × n!
- n! > 2ⁿ for n ≥ 4
- Stirling's Approximation: n! ≈ √(2πn)(n/e)ⁿ
- Double Factorial: n!! = n × (n-2)!!
Applications
- Permutations and Combinations
- Probability Theory
- Taylor Series Expansions
- Graph Theory
- Number Theory
- Quantum Mechanics
Advanced Topics
Permutations (nPr)
Number of ways to arrange r items from n distinct items:
P(n,r) = n!/(n-r)!
Properties:
• P(n,n) = n!
• P(n,1) = n
• P(n,0) = 1
Combinations (nCr)
Number of ways to select r items from n items:
C(n,r) = n!/(r!(n-r)!)
Properties:
• C(n,r) = C(n,n-r)
• C(n,0) = C(n,n) = 1
Gamma Function
Extension of factorial to real and complex numbers:
Γ(n) = (n-1)! for positive integers
Γ(z+1) = z × Γ(z)