1. What is 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. Factorials are fundamental in mathematics, particularly in combinatorics, algebra, and mathematical analysis.

2. How Does the Calculator Work?

The calculator uses the factorial definition:

Special cases:

• \( 0! = 1 \) (by definition)
• \( 1! = 1 \)

3. Applications of Factorial

Details: Factorials are used in:

• Permutations and combinations
• Probability distributions
• Taylor series expansions
• Number theory
• Computational algorithms

4. Using the Calculator

Tips: Enter any non-negative integer up to 170 (higher values exceed floating point precision). The calculator will compute the product of all positive integers up to your input number.

5. Frequently Asked Questions (FAQ)

Q1: Why is 0! equal to 1?
A: This is a convention that makes many mathematical formulas work consistently, particularly in combinatorics where there's exactly one way to arrange zero objects.

Q2: What is the largest factorial this calculator can compute?
A: 170! (approximately 7.26 × 10³⁰⁶), which is near the maximum value representable by floating-point numbers.

Q3: Can factorials be computed for non-integer numbers?
A: For non-integers, the gamma function extends the factorial concept (Γ(n) = (n-1)!), but this calculator only handles integer inputs.

Q4: How are factorials used in probability?
A: They're essential in combinatorics for counting permutations (n!) and combinations (n!/(k!(n-k)!)).

Q5: Why do factorials grow so quickly?
A: Because each multiplication involves an additional term, leading to exponential growth (faster than polynomial but slower than exponential functions like 2ⁿ).