It means 5 factorial, which is 5 × 4 × 3 × 2 × 1 = 120.

Yes, by convention, 0! is defined as 1, which is useful in combinatorial formulas.

Can factorials be calculated for negative numbers or fractions?

Not in standard integer factorial definition. For non-integers, the Gamma function extends the concept.

Where are factorials commonly used?

In permutations, combinations, probability calculations, Taylor series expansions, and algorithm analysis.

factorial

UK/fækˈtɔː.ri.əl/US/fækˈtɔːr.i.əl/

Technical/Academic

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Definition

Meaning

The product of all positive integers from 1 to a given number n, denoted as n!.

In mathematics, the factorial function; also used in combinatorics, probability, and series expansions. Can refer to the mathematical operation itself or the resulting value.

Linguistics

Semantic Notes

Primarily a mathematical term with precise definition. Sometimes used informally in programming/computer science contexts. Not used in everyday conversation outside technical discussions.

Dialectal Variation

British vs American Usage

Differences

No significant differences in meaning or usage. Pronunciation may vary slightly (see IPA).

Connotations

Identical technical connotations in both varieties.

Frequency

Equally rare in general discourse, equally common in mathematical/technical contexts in both regions.

Vocabulary

Collocations

strong

calculate the factorialfactorial of nn factorialfactorial functionfactorial notation

medium

double factorialrising factorialfactorial designfactorial experiment

weak

large factorialcompute factorialfactorial problemfactorial calculation

Grammar

Valency Patterns

the factorial of [number][number] factorialcalculate/compute/find the factorial

Vocabulary

Synonyms

Neutral

n!factorial function

Weak

product seriessequential product

Usage

Context Usage

Business

Virtually never used except in highly technical business analytics or operations research.

Academic

Common in mathematics, statistics, computer science, and engineering publications and lectures.

Everyday

Extremely rare; would only appear in puzzles, advanced schoolwork, or specialist hobbies.

Technical

Core term in mathematics, combinatorics, algorithm analysis, and statistical methods.

Examples

By Part of Speech

adjective

British English

The factorial notation is compact.
We used a factorial design for the experiment.

American English

The factorial expression simplifies the formula.
It's a factorial experiment with two factors.

Examples

By CEFR Level

The factorial of 5 is 120.
In maths, we write 4! for the factorial of four.

You need to understand factorials to solve this permutation problem.
The algorithm's complexity involves calculating a factorial.

Stirling's approximation provides an accurate estimate for large factorials.
The factorial moment generating function is derived from the probability distribution.

Learning

Memory Aids

Mnemonic

Think: 'Factory' produces things; 'factorial' produces the product of a sequence of numbers.

Conceptual Metaphor

A COMPRESSED MULTIPLICATION CHAIN

Watch out

Common Pitfalls

Translation Traps (for Russian speakers)

Do not confuse with 'факториал' (same word, same meaning). No significant trap except ensuring the mathematical concept is understood.

Common Mistakes

Pronouncing as 'factory-al' with equal stress on all syllables. Writing '5!' as '5 factorial' in running text without proper notation can be ambiguous.

Practice

Quiz

Fill in the gap

The number of ways to arrange 7 different books on a shelf is 7 .

Multiple Choice

What is the primary field of use for the word 'factorial'?

FAQ

Frequently Asked Questions

: It means 5 factorial, which is 5 × 4 × 3 × 2 × 1 = 120.
: Yes, by convention, 0! is defined as 1, which is useful in combinatorial formulas.
: Not in standard integer factorial definition. For non-integers, the Gamma function extends the concept.
: In permutations, combinations, probability calculations, Taylor series expansions, and algorithm analysis.