Factorial

The factorial n! = n×(n-1)×...×1 is the basis of permutations and combinations; it grows extremely fast, with 20! already exceeding the 64-bit integer range.

Purpose

To count permutations of n elements, a foundation of combinatorics.

Mechanism

n! = n×(n-1)!, with 0!=1. Stirling's approximation: n! ≈ √(2πn)(n/e)^n. Large n requires BigInteger. The Γ function extends the factorial to non-integers. Log-factorials use lgamma to avoid overflow.

Use cases

Permutation and combination counting
Probability and statistics (Poisson distribution)
Taylor series coefficients
Gamma function

Examples

Input
```
5=5×4×3×2×1=120
```
Output
```
120
```
Number of permutations of 5 elements

Common errors

Factorial overflow for large n: Use BigInteger or log-factorials

Related terms

A permutation counts the ordered ways to choose k elements from n, P(n,k)=n!/(n-k)!; the full permutation count is n!.

A combination counts the ways to choose k elements from n without regard to order, C(n,k)=n!/(k!(n-k)!), i.e., the binomial coefficient.

Fibonacci Sequence

The Fibonacci sequence has each term equal to the sum of the two before it (0,1,1,2,3,5,8...); it relates to the golden ratio and appears widely in nature and algorithms.

BigInteger is an arbitrary-precision integer type, not limited by a fixed word size, used for computations beyond the native integer range (such as cryptography and combinatorics).

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References