/Factorial-

Factorial-

Factorial:

Definition: Factorial is a mathematical operation denoted by the symbol "!" that represents the product of all positive integers less than or equal to a given positive integer (n). It is often expressed as (n!), where (n) is a non-negative integer. Mathematically, (n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1).

Algorithm:

  1. The factorial function takes a non-negative integer n as input.
  2. If n is 0 or 1, the function returns 1 because the factorial of 0 and 1 is 1.
  3. If n is greater than 1, the function initializes a variable result to 1. It then uses a for loop to iterate from 2 to n (inclusive) and multiplies each number to the result.
  4. The final result is the factorial of the input number.

Uses of Factorial:

  1. Combinatorics:

    • Factorials are used in combinatorics to calculate permutations and combinations.
    • For example, the number of ways to arrange (n) distinct items is given by (n!).

  2. Probability:

    • In probability theory, factorials are used to calculate the number of possible outcomes in a sample space.

  3. Series and Calculus:

    • Factorials are used in the series expansion of functions such as the exponential function and binomial theorem.

  4. Recursive Definitions:

    • Factorials often appear in recursive definitions of mathematical and computational concepts.

  5. Mathematical Identities:

    • Factorials are involved in various mathematical identities and formulas.

In summary, the factorial of a number is a crucial mathematical operation with applications in combinatorics, probability, series expansion, and various mathematical concepts. In programming, calculating factorials is a common task and is often implemented using loops or recursion.