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
Key Notes
- Definition: n! = n × (n−1) × (n−2) × … × 1: For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. The special case 0! = 1 is defined by convention — it ensures that combinatorial formulas C(n, 0) = 1 and C(n, n) = 1 work correctly.
- Superexponential growth: Factorials grow faster than any exponential. 10! = 3,628,800; 20! ≈ 2.4 × 10¹⁸; 52! (card deck arrangements) ≈ 8 × 10⁶⁷ — vastly larger than the number of atoms in the observable universe.
- Stirling's approximation: n! ≈ √(2πn) × (n/e)^n: Essential when n is too large for direct computation. For n = 100: log₁₀(100!) ≈ 157.97. The approximation is accurate to within 1% for n ≥ 10.
- Building block of combinatorics: Combinations C(n,k) = n! / [k!(n−k)!] count selections where order doesn't matter. Permutations P(n,k) = n! / (n−k)! count ordered arrangements. Both are derived directly from factorial counting of arrangements.
- Applications: Factorials appear in Taylor series coefficients (e.g., e^x = Σ xⁿ/n!), probability distributions (binomial, Poisson), counting algorithms, and cryptography (estimating brute-force complexity of permutation ciphers).