Factorials: n! and Its Explosive Growth
The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n. Factorials are the building block of combinatorics — counting arrangements, combinations, and probabilities.
Definition
n! = n × (n−1) × (n−2) × ... × 2 × 1
0! = 1 (by definition)
1! = 1 | 2! = 2 | 3! = 6 | 4! = 24 | 5! = 120
6! = 720 | 7! = 5,040 | 10! = 3,628,800
20! ≈ 2.43 × 10¹⁸Why 0! = 1
There is exactly one way to arrange zero objects (do nothing). This preserves the recursive relationship n! = n × (n−1)! and makes combinatorial formulas work correctly.
Permutations and Combinations
Permutations (order matters): P(n,r) = n! ÷ (n−r)!
Combinations (order irrelevant): C(n,r) = n! ÷ [r!(n−r)!]
Example: Choose 3 from 5:
P(5,3) = 120 ÷ 2 = 60
C(5,3) = 120 ÷ (6×2) = 10Stirling's Approximation (Large n)
n! ≈ √(2πn) × (n/e)ⁿ
100! ≈ 9.33 × 10¹⁵⁷Calculate factorials: Free Factorial Calculator
Factorial Values
- 0! = 1 (by definition)
- 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, 12! = 479,001,600, 15! ≈ 1.307 × 10¹²
- 20! ≈ 2.432 × 10¹⁸, 52! ≈ 8.066 × 10⁶⁷ (card shuffles)
Applications in Counting
Factorials are the foundation of combinatorics — the mathematics of counting arrangements and selections. Permutations (ordered arrangements): P(n,r) = n!/(n-r)! counts ways to choose r items from n in order. Combinations (unordered selections): C(n,r) = n!/(r!(n-r)!) counts ways to choose r from n without regard to order. Example: choosing a 5-card poker hand from a 52-card deck = C(52,5) = 52!/(5!·47!) = 2,598,960 possible hands. The number of ways to shuffle a standard deck is 52! ≈ 8 × 10⁶⁷ — more than the estimated number of atoms in the observable universe.
Frequently Asked Questions
Why is 0! = 1?
There is exactly one way to arrange zero objects: do nothing. This is the empty arrangement. Defining 0! = 1 also ensures that combination formulas work correctly: C(n,0) = n!/(0!·n!) = 1, meaning there is exactly one way to choose nothing from a set — which is true. Without this definition, many formulas in combinatorics would require special cases.
How fast do factorials grow?
Extremely fast — faster than any polynomial or exponential function. 13! already exceeds one billion (6.2 × 10⁹). 21! exceeds the number of seconds since the Big Bang. For large n, Stirling's approximation provides a practical estimate: n! ≈ √(2πn) × (n/e)ⁿ.
Are factorials defined for non-integers?
Yes, through the Gamma function: Γ(n) = (n-1)! for positive integers, but Γ is defined for all positive real numbers. For example, Γ(1/2) = √π, which means (-1/2)! = √π. This generalisation is used in probability distributions (gamma distribution, chi-squared distribution) and in advanced integration.