Factorialis

Factorialis is a mathematical function that takes a non-negative integer as input and returns the product of all positive integers less than or equal to that integer. It is denoted by an exclamation mark (!) placed after the integer. For example, 5 factorial (written as 5!) is calculated as 5 4 3 2 1, which equals 120.

The factorial of zero (0!) is defined as 1. This definition is important for various mathematical formulas and combinatorial calculations. The factorial function grows very rapidly, meaning that even for relatively small input numbers, the output can become extremely large.

Factorials are fundamental in combinatorics, particularly in calculating permutations and combinations. A permutation is an arrangement of objects in a specific order, and the number of permutations of n distinct objects is n!. A combination refers to the selection of objects where the order does not matter, and factorial functions are used in the formula to calculate the number of combinations.

The factorial function can also be extended to non-integer values using the Gamma function, which is a generalization of the factorial. However, in its basic definition, factorial is strictly for non-negative integers.