Skip to main content
Select Source:

Factorial

Factorial


The pattern of multiplying a positive integer by the next lower consecutive integer occurs frequently in mathematics. Look for the pattern in the following expressions.

7 × 6 × 5 × 4 × 3 × 2 × 1

4 × 3 × 2 × 1

(n + 5) × (n + 4) × (n + 3) × (n + 2) × (n + 1) × n

The mathematical symbol for this string of factors is the familiar exclamation point (!). This pattern of multiplied whole numbers is called n factorial and is written as n ! So, starting with the greatest factor, n, the factorial pattern is as follows:

n ! = n (n - 1)(n - 2)(n - 3) (1).

So,

3! is 3 × 2 × 1 = 6

5! is 5 × 4 × 3 × 2 × 1 = 120 and 1! = 1.

Zero factorial (0!) is arbitrarily defined to be 1.

Most scientific calculators have a key (such as x !) that can be used to find factorial values. As n becomes larger, the value of its factorials increases rapidly. For example, 13! is 6,227,020,800.

How Factorials Are Used

Many mathematical formulas use factorial notation, including the formulas for finding permutations and combinations . For example, the number of permutations of n elements taken n at a time is n !, and the number of permutations of n elements taken r at a time is equal to .

There is also a problem that involves prime and composite numbers which uses a formula containing factorial notation. Mathematicians have, for many years, puzzled over the question of how prime numbers were distributed. Notice that, in the whole numbers less than 20, there are eight prime numbers (2, 3, 5, 7, 11, 13, 17, and 19). But from 20 to 40, there are only four prime numbers (23, 29, 31, and 37).

No one has yet found a formula that will generate all the prime numbers. However, the following sequence will give a string of n consecutive composite numbers (numbers that are not prime) for any positive integer n.

(n + 1)! + 2, (n + 1)! + 3, (n + 1)! + 4, (n + 1)! + 5, (n + 1)! + 6, and so on up to (n + 1)! + (n + 1).

When n is 2, notice that this sequence only has two terms:

(n + 1)! + 2, (n + 1)! + (n + 1)

which is

(2 + 1)! + 2, (2 + 1)! + (2 + 1)

For the first term, (2 + 1)! + 2 is 3! + 2 or (3 × 2 × 1) + 2, giving a value of 8. The second term has a value of 9.

When n = 2, this sequence gives two consecutive numbers that are not prime numbers: 8, 9. When n = 3, this sequence gives three consecutive numbers that are not prime numbers: 26, 27, 28. This relationship between the value of n and the number of consecutive numbers that are not prime numbers continues in this sequence for any whole number value for n. For a greater n, such as 300, a sequence of 300 composite numbers (that is, a list of 300 consecutive numbers with no prime number in the list) can be found.

see also Factors; Primes, Puzzles of; Permutations and Combinations.

Lucia McKay

Bibliography

Stephens, Larry. Algebra for the Utterly Confused. New York: McGraw-Hill, 2000.

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"Factorial." Mathematics. . Encyclopedia.com. 23 May. 2018 <http://www.encyclopedia.com>.

"Factorial." Mathematics. . Encyclopedia.com. (May 23, 2018). http://www.encyclopedia.com/education/news-wires-white-papers-and-books/factorial

"Factorial." Mathematics. . Retrieved May 23, 2018 from Encyclopedia.com: http://www.encyclopedia.com/education/news-wires-white-papers-and-books/factorial

Learn more about citation styles

Citation styles

Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

http://www.mla.org/style

The Chicago Manual of Style

http://www.chicagomanualofstyle.org/tools_citationguide.html

American Psychological Association

http://apastyle.apa.org/

Notes:
  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.

factorial

fac·to·ri·al / fakˈtôrēəl/ • n. Math. the product of an integer and all the integers below it; e.g., factorial four (4!) is equal to 24. (Symbol: !) ∎  the product of a series of factors in an arithmetic progression. • adj. chiefly Math. relating to a factor or such a product: a factorial design. DERIVATIVES: fac·to·ri·al·ly adv.

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"factorial." The Oxford Pocket Dictionary of Current English. . Encyclopedia.com. 23 May. 2018 <http://www.encyclopedia.com>.

"factorial." The Oxford Pocket Dictionary of Current English. . Encyclopedia.com. (May 23, 2018). http://www.encyclopedia.com/humanities/dictionaries-thesauruses-pictures-and-press-releases/factorial

"factorial." The Oxford Pocket Dictionary of Current English. . Retrieved May 23, 2018 from Encyclopedia.com: http://www.encyclopedia.com/humanities/dictionaries-thesauruses-pictures-and-press-releases/factorial

Learn more about citation styles

Citation styles

Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

http://www.mla.org/style

The Chicago Manual of Style

http://www.chicagomanualofstyle.org/tools_citationguide.html

American Psychological Association

http://apastyle.apa.org/

Notes:
  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.