Calculate factorials (n!) for permutations, combinations, and probability
A factorial, denoted by n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental in mathematics, particularly in counting, probability, and combinatorics.
The factorial function grows extremely rapidly. While 5! = 120, by the time you reach 10! you get 3,628,800, and 20! exceeds 2 quintillion. This rapid growth makes factorials important in complexity analysis and algorithm design.
By convention, 0! is defined as 1. This isn't intuitive, but it's necessary for many mathematical formulas to work correctly, especially in combinatorics where choosing 0 items from a set should give exactly 1 way (choosing nothing).
P(n,r) = n!/(n-r)! counts arrangements where order matters.
C(n,r) = n!/[r!(n-r)!] counts selections where order doesn't matter.
Used in binomial distribution, Poisson distribution, and other probability calculations.
Factorials appear in denominators of Taylor series expansions for sin, cos, e^x.
Here are the first 15 factorial values for quick reference:
| n | n! | Digits | Uses |
|---|---|---|---|
| 0 | 1 | 1 | By definition |
| 1 | 1 | 1 | Base case |
| 5 | 120 | 3 | Hand arrangements |
| 7 | 5,040 | 4 | Week permutations |
| 10 | 3,628,800 | 7 | Digit arrangements |
| 12 | 479,001,600 | 9 | Month permutations |
| 15 | 1,307,674,368,000 | 13 | Large calculations |
n! = n × (n-1)!. This property is useful for computing factorials programmatically and understanding factorial relationships.
For large n: n! ≈ √(2πn)(n/e)^n. This approximation is useful when exact factorials are too large to compute.
The number of trailing zeros in n! equals floor(n/5) + floor(n/25) + floor(n/125) + ... This counts how many times 10 divides n!.
Factorials extend to non-integers via Γ(n+1) = n!. The gamma function allows factorial-like calculations for any positive real number.
By convention and for mathematical consistency. The empty product (multiplying no numbers) is 1, and this makes formulas like C(n,0) = n!/(0!×n!) = 1 work correctly.
No, factorials are only defined for non-negative integers. However, the gamma function Γ(n) extends the concept to complex numbers except negative integers.
Most calculators overflow around 170! which exceeds 10^300. This calculator uses special techniques to display larger factorials as scientific notation.
Double factorial n!! means multiplying every other number: 7!! = 7×5×3×1 = 105. It's not (n!)! but a separate operation used in combinatorics.
You want to know how many distinct ways 7 different books can be ordered on a shelf — a classic permutation problem solved with a factorial.
Result7! = 5,040 arrangements
Multiply 7 × 6 × 5 × 4 × 3 × 2 × 1. There are 7 choices for the first slot, 6 for the second, and so on down to 1. The product is 5,040, the same value that appears inside permutation formulas like P(n, r) = n! / (n − r)!.
0! = 1 is a convention chosen so that core formulas stay consistent. The empty product of no factors is 1, and definitions like C(n, 0) = n! / (0! · n!) = 1 and the Taylor series 1 + x/1! + x²/2! + … only work if 0! equals 1.
Factorials appear throughout probability, combinatorics, and calculus. Permutations P(n, r) = n! / (n − r)! and combinations C(n, r) = n! / [r!(n − r)!] count arrangements and selections, the binomial and Poisson distributions use them, and Taylor series for e^x, sin x, and cos x divide each term by k!.
Stirling's approximation says n! ≈ √(2πn) · (n/e)^n for large n. It is useful when exact factorials are too large to store or when you only need orders of magnitude, such as in statistical mechanics, information theory, or analyzing algorithm complexity. The relative error shrinks as n grows.
The gamma function Γ(z) is defined by an integral and satisfies Γ(n + 1) = n! for non-negative integers n. It extends factorials to all complex numbers except the non-positive integers, which lets you compute values like (1/2)! = √π / 2 used in statistics and physics.
The double factorial n!! is the product of every other integer down to 1 or 2. For example, 7!! = 7 × 5 × 3 × 1 = 105 and 8!! = 8 × 6 × 4 × 2 = 384. It is not (n!)!, which would be a factorial of a factorial. Double factorials appear in formulas for the volume of n-balls and certain trigonometric integrals.
Faster than any exponential. 5! = 120, 10! = 3,628,800, and 20! ≈ 2.43 × 10^18. By 70! the result exceeds 10^100 (a googol), and 170! is near the upper limit of standard double-precision floating point at roughly 10^306.
No. The factorial is only defined for non-negative integers. The gamma function extends the idea to other numbers, but Γ(z) has simple poles at z = 0, −1, −2, …, so negative integers still have no finite factorial value.
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