Factorial Formula
Calculate n factorial (n!) — the product of all positive integers from 1 to n.
Used in permutations, combinations, and probability.
The Formula
n! = n × (n - 1) × (n - 2) × ... × 2 × 1
The factorial of a number n is the product of all positive integers from 1 up to n.
By definition, 0! = 1.
Variables
| Symbol | Meaning |
|---|---|
| n | A non-negative integer |
| n! | n factorial — the product of all integers from 1 to n |
Common Factorial Values
- 0! = 1 (by definition)
- 1! = 1
- 2! = 2
- 3! = 6
- 4! = 24
- 5! = 120
- 6! = 720
- 7! = 5,040
- 10! = 3,628,800
Example 1
Calculate 6!
6! = 6 × 5 × 4 × 3 × 2 × 1
6! = 30 × 4 × 3 × 2 × 1 = 120 × 3 × 2 × 1 = 360 × 2 × 1
6! = 720
Example 2
Simplify 8! / 6!
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
6! = 6 × 5 × 4 × 3 × 2 × 1
8! / 6! = (8 × 7 × 6!) / 6! = 8 × 7
8! / 6! = 56
When to Use It
Use the factorial formula when:
- Counting the number of ways to arrange items in order (permutations)
- Calculating combinations and probability
- Working with the binomial theorem
- Solving counting problems in combinatorics