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. By definition, 0! = 1.

2. How Factorial Calculation Works

The factorial is calculated using the formula:

Example: 5! = 5 × 4 × 3 × 2 × 1 = 120

3. Applications of Factorial

Details: Factorials are used extensively in mathematics, particularly in combinatorics, algebra, and mathematical analysis. They appear in permutations, combinations, Taylor series, and probability calculations.

4. Using the Calculator

Tips: Enter any integer between 0 and 170. The calculator will compute the factorial value. Note that factorials grow very rapidly, so values above 170 exceed typical computational limits.

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: The calculator can compute up to 170! (approximately 7.26 × 10³⁰⁶). Beyond this, numbers exceed PHP's floating point maximum.

Q3: Can factorials be computed for negative numbers?
A: No, factorials are only defined for non-negative integers. However, the gamma function extends the concept to complex numbers.

Q4: How are factorials used in probability?
A: Factorials count permutations (ordered arrangements) and combinations (unordered selections) of items, fundamental in probability calculations.

Q5: What's the relationship between factorials and binomial coefficients?
A: Binomial coefficients (n choose k) are calculated as n!/(k!(n-k)!), representing combinations.