Factorials are mathematical functions commonly denoted by an exclamation mark (!). They are used to represent the product of a whole number and all the positive integers below it. Specifically, a factorial of a non-negative integer n, denoted as n!, is the product of all positive integers from 1 to n. For instance, 5! would equal 5 × 4 × 3 × 2 × 1 = 120.
Factorials often appear in various branches of mathematics, especially in combinatorics and probability theory. They have applications in counting permutations, arranging objects, calculating probability outcomes, and solving equations involving permutations or combinations. Factorials also emerge in calculus and in solving equations with factorial terms, known as factorial equations.
Factorials possess several important properties, such as being defined for non-negative integers only. Zero factorial, denoted as 0!, is considered to be equal to 1 as there is only one way to arrange zero objects: by doing nothing. It is crucial to note that factorials grow rapidly as the input number increases. For instance, the factorial of 10 is 3,628,800, while the factorial of 20 is 2,432,902,008,176,640,000.
In summary, factorials are mathematical functions used to express the product of a whole number and all positive integers below it. They find applications in various mathematical fields, particularly in counting arrangements and solving equations.
