# factorial

## English

### Pronunciation

• IPA(key): /fækˈtɔːɹi.əl/
•  Audio (Southern England) (file)

### Noun

factorial (plural factorials)

1. The result of multiplying a given number of consecutive integers from 1 to the given number. In equations, it is symbolized by an exclamation mark (!). For example, 5! = 1 × 2 × 3 × 4 × 5 = 120.
• 1916 July, M. Mott-Smith, “The Arithmetical Pyramid of Many Dimensions”, in The Monist, volume 26, number 3, →DOI, page 428:
The expression ${}_{n}C_{r}$  means the number of combinations of $n$  things taken $r$  at a time. It is also written ${\binom {n}{r}}$ , and is equal to
${\frac {n(n-1)(n-2)(n-3)\cdots \cdots (n-r+1)}{r!}}$

or $n!/r!(n-r)!$ , in which $r!$  is read "factorial $r$ " and denotes the product of all the integral numbers from $1$  to $r$  inclusive.
• 2018 September, Colin Foster, “What is the formula for factorial?”, in Mathematics in School, volume 47, number 4, page 40:
This playing around with consecutive integers reminded one of the students of factorials, and he asked about products of integers, which I said could be expressed similarly as
$\prod _{k=1}^{n}k=1.2.3\cdots n.$

But what does this equal?
Well, of course we can say that it is equal to $n$  factorial and write
$\prod _{k=1}^{n}k=n!$

and then use our calculators, but inventing a new symbol like 'factorial' feels like an admission of defeat!

#### Usage notes

"n!" is read as "factorial of n" or "n factorial."

#### Translations

factorial (comparative more factorial, superlative most factorial)

1. Of or pertaining to a factor or factorial.
• 1903, Arthur Schuster, “On some Definite Integrals, and a New Method of Reducing a Function of Spherical Co-ordintes to a Series of Spherical Harmonics”, in Philosophical Transactions of the Royal Society A, volume 200, →DOI, →JSTOR, page 182:
I shall denote, as usual, the factorial product of the numbers up to $n$  by $n!$ , but I have found it necessary to introduce a separate notation for the products of successive even or odd numbers. I consequently define
$n!!=n.(n-2)!!.$

Starting with
$1!!=1,\quad 2!!=2,$

it follows that, for positive values of $n$ ,
$n!!=n\cdot n-2\cdot n-4\cdots ,$

where the last factor is either $1$  or $2$ , according as $n$  is odd or even.
• 1995, K. E. Hirst, Numbers, Sequences and Series, Butterworth-Heinemann, →ISBN, page 21:
The definition of addition is an example of definition by induction, sometimes called recursive definition. As another example of this, consider the following inductive definition of the factorial function.
We define $n!$  for $n\in \mathbb {N}$  by the following two properties,
(i) $1!$  is defined to be $1$ ,
(ii) for all $k\in \mathbb {N}$ , we define $(k+1)!$  to be $(k+1)\times k!$ .
2. Of or pertaining to a factor, a kind of business agent.
• 2004, The Digest: Annotated British, Commonwealth, and European Cases:
The latter sold the goods to a customer who was cashier to certain creditors of the agents without disclosing the factorial capacity in which they acted.
3. (dated) Of or pertaining to a factory.

## Portuguese

### Noun

factorial m (plural factoriais)

1. Pre-reform spelling (until Brazil 1943/Portugal 1990) of fatorial. Still used in countries where the agreement hasn't come into effect; may occur as a sporadic misspelling.

factorial m or f (plural factoriais)

1. Pre-reform spelling (until Brazil 1943/Portugal 1990) of fatorial. Still used in countries where the agreement hasn't come into effect; may occur as a sporadic misspelling.

## Romanian

### Etymology

factorial m or n (feminine singular factorială, masculine plural factoriali, feminine and neuter plural factoriale)

1. factorial

## Spanish

### Pronunciation

• IPA(key): /faɡtoˈɾjal/ [faɣ̞.t̪oˈɾjal]
• Rhymes: -al
• Syllabification: fac‧to‧rial

factorial m or f (masculine and feminine plural factoriales)

1. factorial

### Noun

factorial m (plural factoriales)

1. factorial