Faktoriaalinen

Faktoriaalinen is a mathematical operation denoted by an exclamation mark (!). For a non-negative integer n, the factorial of n, written as n!, is the product of all positive integers less than or equal to n. For example, 5! is calculated as 5 × 4 × 3 × 2 × 1, which equals 120. The factorial of 0 is defined as 1 (0! = 1). This convention is useful in various mathematical contexts, particularly in combinatorics and calculus.

Factorials grow very rapidly. For instance, 10! is 3,628,800, and 20! is a much larger number, approximately 2.43 × 10^18. The factorial function is not defined for negative integers. For non-integer values, the gamma function is a generalization of the factorial.

Factorials appear in many areas of mathematics. They are fundamental in calculating permutations and combinations. For instance, the number of ways to arrange n distinct items is n!. The number of ways to choose k items from a set of n items (combinations) is given by the binomial coefficient, which is expressed using factorials: n! / (k! * (n-k)!). They also feature in Taylor series expansions of functions like e^x and sin(x).