Understanding Factorials

What is a 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. For example:

5! = 5 × 4 × 3 × 2 × 1 = 120
4! = 4 × 3 × 2 × 1 = 24
3! = 3 × 2 × 1 = 6
2! = 2 × 1 = 2
1! = 1
0! = 1 (by definition)

Applications of Factorials

Factorials are essential in various mathematical domains:

Combinatorics: Calculating permutations and combinations
Probability: Determining possible outcomes in events
Calculus: Taylor series expansions
Algebra: Solving problems involving sequences and series

Factorial Reference Table

n	n!
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5,040
8	40,320
9	362,880
10	3,628,800

Calculation Tips

Factorials grow extremely rapidly - 10! is already 3.6 million
For values above 20, we use Stirling's approximation: n! ≈ √(2πn) × (n/e)^n
Negative numbers don't have factorial values (they have gamma function extensions)
Decimal values require the gamma function Γ(z) which extends factorials to complex numbers

Frequently Asked Questions

Why is 0! equal to 1?

This is a convention that maintains consistency in mathematical formulas, particularly in combinatorics where there is exactly one way to arrange zero objects - the empty arrangement.

Can factorials be calculated for negative numbers?

No, factorials are only defined for non-negative integers. However, the gamma function extends the concept to complex numbers (except negative integers).

What is the largest factorial this calculator can compute?

For exact calculation, we can compute up to 20! (2,432,902,008,176,640,000). Beyond that, we use Stirling's approximation which provides close estimates for larger values.

What are nCr and nPr?

nCr (combinations) represents the number of ways to choose r items from n items without regard to order.
nPr (permutations) represents the number of ways to arrange r items from n items where order matters.

Factorial Calculator

Result

Related Calculations

Menu

Factorial Calculator - n Factorial Fast

AI Assistant

Was this tool helpful?