factorial

/fæk'tɔ:riəl/

Thân thiện

A student calculates the factorial of 5 on a chalkboard.

Definition

1. Noun:
   • The product of all positive integers up to and including a given integer: In mathematics, the "factorial" of a non-negative integer n is the product of all positive integers less than or equal to n. It is denoted by the symbol n!.
2. Adjective:
   • Of or relating to factorials: Describes something that involves or pertains to the mathematical operation of a factorial.

Usage

• Noun: The factorial is a fundamental function in combinatorics, probability, and algebra. It is used to calculate permutations and combinations.
• Adjective: The term is used to describe calculations, properties, or notations related to this specific product.

Examples

• Noun:
  • The factorial of 5 (5!) is 1 × 2 × 3 × 4 × 5 = 120.
  • In how many ways can 4 books be arranged on a shelf? The answer is the factorial of 4.
• Adjective:
  • We used a factorial notation to simplify the equation.
  • The growth of the factorial function is extremely rapid.

Advanced Usage

• "Double factorial" (n!!): The product of all the integers from 1 up to that have the same parity (odd or even) as . For example, 7!! = 1 × 3 × 5 × 7 = 105.
• "Factorial prime": A prime number that is one more or one less than a factorial. For example, 7 is a factorial prime because 3! + 1 = 7.

Variants and Related Words

• Factorialize (verb, rare): To compute the factorial of a number.
• Subfactorial (noun, !n): The number of derangements (permutations where no element appears in its original position) of elements.

Synonyms

• Noun: (the standard symbolic notation).
• Adjective: Combinatorial (in some specific contexts, though not a direct synonym).

Related Phrases & Concepts

• Gamma function: A complex function that extends the factorial to real and complex numbers.
• Stirling's approximation: A formula for approximating the value of large factorials.
• Binomial coefficient: A number expressed using factorials: .

Học thuật

Adjective

1. of or relating to factorials

Noun

1. the product of all the integers up to and including a given integer
   • 1, 2, 6, 24, and 120 are factorials